1 \input texinfo @c -*-texinfo-*-
3 @setfilename ginac.info
4 @settitle GiNaC, an open framework for symbolic computation within the C++ programming language
11 @c I hate putting "@noindent" in front of every paragraph.
19 * ginac: (ginac). C++ library for symbolic computation.
23 This is a tutorial that documents GiNaC @value{VERSION}, an open
24 framework for symbolic computation within the C++ programming language.
26 Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author The GiNaC Group:
51 @author Christian Bauer, Alexander Frink, Richard Kreckel, Jens Vollinga
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2005 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A Tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic Concepts:: Description of fundamental classes.
85 * Methods and Functions:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A Comparison With Other CAS:: Compares GiNaC to traditional CAS.
88 * Internal Structures:: Description of some internal structures.
89 * Package Tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A Tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistic structures that allow for the creation of large-scale
105 projects. GiNaC is an attempt to overcome this situation by extending a
106 well established and standardized computer language (C++) by some
107 fundamental symbolic capabilities, thus allowing for integrated systems
108 that embed symbolic manipulations together with more established areas
109 of computer science (like computation-intense numeric applications,
110 graphical interfaces, etc.) under one roof.
112 The particular problem that led to the writing of the GiNaC framework is
113 still a very active field of research, namely the calculation of higher
114 order corrections to elementary particle interactions. There,
115 theoretical physicists are interested in matching present day theories
116 against experiments taking place at particle accelerators. The
117 computations involved are so complex they call for a combined symbolical
118 and numerical approach. This turned out to be quite difficult to
119 accomplish with the present day CAS we have worked with so far and so we
120 tried to fill the gap by writing GiNaC. But of course its applications
121 are in no way restricted to theoretical physics.
123 This tutorial is intended for the novice user who is new to GiNaC but
124 already has some background in C++ programming. However, since a
125 hand-made documentation like this one is difficult to keep in sync with
126 the development, the actual documentation is inside the sources in the
127 form of comments. That documentation may be parsed by one of the many
128 Javadoc-like documentation systems. If you fail at generating it you
129 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
130 page}. It is an invaluable resource not only for the advanced user who
131 wishes to extend the system (or chase bugs) but for everybody who wants
132 to comprehend the inner workings of GiNaC. This little tutorial on the
133 other hand only covers the basic things that are unlikely to change in
137 The GiNaC framework for symbolic computation within the C++ programming
138 language is Copyright @copyright{} 1999-2005 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
157 @node A Tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A Tour of GiNaC, A Tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lcln -lginac
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package Tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A Tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected and normalized (i.e. converted to a ratio of two coprime
379 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
380 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
381 > b = x^2 + 4*x*y - y^2;
384 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
386 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
388 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
393 You can differentiate functions and expand them as Taylor or Laurent
394 series in a very natural syntax (the second argument of @code{series} is
395 a relation defining the evaluation point, the third specifies the
398 @cindex Zeta function
402 > series(sin(x),x==0,4);
404 > series(1/tan(x),x==0,4);
405 x^(-1)-1/3*x+Order(x^2)
406 > series(tgamma(x),x==0,3);
407 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
408 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
410 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
411 -(0.90747907608088628905)*x^2+Order(x^3)
412 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
413 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
414 -Euler-1/12+Order((x-1/2*Pi)^3)
417 Here we have made use of the @command{ginsh}-command @code{%} to pop the
418 previously evaluated element from @command{ginsh}'s internal stack.
420 Often, functions don't have roots in closed form. Nevertheless, it's
421 quite easy to compute a solution numerically, to arbitrary precision:
426 > fsolve(cos(x)==x,x,0,2);
427 0.7390851332151606416553120876738734040134117589007574649658
429 > X=fsolve(f,x,-10,10);
430 2.2191071489137460325957851882042901681753665565320678854155
432 -6.372367644529809108115521591070847222364418220770475144296E-58
435 Notice how the final result above differs slightly from zero by about
436 @math{6*10^(-58)}. This is because with 50 decimal digits precision the
437 root cannot be represented more accurately than @code{X}. Such
438 inaccuracies are to be expected when computing with finite floating
441 If you ever wanted to convert units in C or C++ and found this is
442 cumbersome, here is the solution. Symbolic types can always be used as
443 tags for different types of objects. Converting from wrong units to the
444 metric system is now easy:
452 140613.91592783185568*kg*m^(-2)
456 @node Installation, Prerequisites, What it can do for you, Top
457 @c node-name, next, previous, up
458 @chapter Installation
461 GiNaC's installation follows the spirit of most GNU software. It is
462 easily installed on your system by three steps: configuration, build,
466 * Prerequisites:: Packages upon which GiNaC depends.
467 * Configuration:: How to configure GiNaC.
468 * Building GiNaC:: How to compile GiNaC.
469 * Installing GiNaC:: How to install GiNaC on your system.
473 @node Prerequisites, Configuration, Installation, Installation
474 @c node-name, next, previous, up
475 @section Prerequisites
477 In order to install GiNaC on your system, some prerequisites need to be
478 met. First of all, you need to have a C++-compiler adhering to the
479 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
480 so if you have a different compiler you are on your own. For the
481 configuration to succeed you need a Posix compliant shell installed in
482 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
483 process as well, since some of the source files are automatically
484 generated by Perl scripts. Last but not least, Bruno Haible's library
485 CLN is extensively used and needs to be installed on your system.
486 Please get it either from @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
487 @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
488 from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
489 site} (it is covered by GPL) and install it prior to trying to install
490 GiNaC. The configure script checks if it can find it and if it cannot
491 it will refuse to continue.
494 @node Configuration, Building GiNaC, Prerequisites, Installation
495 @c node-name, next, previous, up
496 @section Configuration
497 @cindex configuration
500 To configure GiNaC means to prepare the source distribution for
501 building. It is done via a shell script called @command{configure} that
502 is shipped with the sources and was originally generated by GNU
503 Autoconf. Since a configure script generated by GNU Autoconf never
504 prompts, all customization must be done either via command line
505 parameters or environment variables. It accepts a list of parameters,
506 the complete set of which can be listed by calling it with the
507 @option{--help} option. The most important ones will be shortly
508 described in what follows:
513 @option{--disable-shared}: When given, this option switches off the
514 build of a shared library, i.e. a @file{.so} file. This may be convenient
515 when developing because it considerably speeds up compilation.
518 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
519 and headers are installed. It defaults to @file{/usr/local} which means
520 that the library is installed in the directory @file{/usr/local/lib},
521 the header files in @file{/usr/local/include/ginac} and the documentation
522 (like this one) into @file{/usr/local/share/doc/GiNaC}.
525 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
526 the library installed in some other directory than
527 @file{@var{PREFIX}/lib/}.
530 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
531 to have the header files installed in some other directory than
532 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
533 @option{--includedir=/usr/include} you will end up with the header files
534 sitting in the directory @file{/usr/include/ginac/}. Note that the
535 subdirectory @file{ginac} is enforced by this process in order to
536 keep the header files separated from others. This avoids some
537 clashes and allows for an easier deinstallation of GiNaC. This ought
538 to be considered A Good Thing (tm).
541 @option{--datadir=@var{DATADIR}}: This option may be given in case you
542 want to have the documentation installed in some other directory than
543 @file{@var{PREFIX}/share/doc/GiNaC/}.
547 In addition, you may specify some environment variables. @env{CXX}
548 holds the path and the name of the C++ compiler in case you want to
549 override the default in your path. (The @command{configure} script
550 searches your path for @command{c++}, @command{g++}, @command{gcc},
551 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
552 be very useful to define some compiler flags with the @env{CXXFLAGS}
553 environment variable, like optimization, debugging information and
554 warning levels. If omitted, it defaults to @option{-g
555 -O2}.@footnote{The @command{configure} script is itself generated from
556 the file @file{configure.ac}. It is only distributed in packaged
557 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
558 must generate @command{configure} along with the various
559 @file{Makefile.in} by using the @command{autogen.sh} script. This will
560 require a fair amount of support from your local toolchain, though.}
562 The whole process is illustrated in the following two
563 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
564 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
567 Here is a simple configuration for a site-wide GiNaC library assuming
568 everything is in default paths:
571 $ export CXXFLAGS="-Wall -O2"
575 And here is a configuration for a private static GiNaC library with
576 several components sitting in custom places (site-wide GCC and private
577 CLN). The compiler is persuaded to be picky and full assertions and
578 debugging information are switched on:
581 $ export CXX=/usr/local/gnu/bin/c++
582 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
583 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
584 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
585 $ ./configure --disable-shared --prefix=$(HOME)
589 @node Building GiNaC, Installing GiNaC, Configuration, Installation
590 @c node-name, next, previous, up
591 @section Building GiNaC
592 @cindex building GiNaC
594 After proper configuration you should just build the whole
599 at the command prompt and go for a cup of coffee. The exact time it
600 takes to compile GiNaC depends not only on the speed of your machines
601 but also on other parameters, for instance what value for @env{CXXFLAGS}
602 you entered. Optimization may be very time-consuming.
604 Just to make sure GiNaC works properly you may run a collection of
605 regression tests by typing
611 This will compile some sample programs, run them and check the output
612 for correctness. The regression tests fall in three categories. First,
613 the so called @emph{exams} are performed, simple tests where some
614 predefined input is evaluated (like a pupils' exam). Second, the
615 @emph{checks} test the coherence of results among each other with
616 possible random input. Third, some @emph{timings} are performed, which
617 benchmark some predefined problems with different sizes and display the
618 CPU time used in seconds. Each individual test should return a message
619 @samp{passed}. This is mostly intended to be a QA-check if something
620 was broken during development, not a sanity check of your system. Some
621 of the tests in sections @emph{checks} and @emph{timings} may require
622 insane amounts of memory and CPU time. Feel free to kill them if your
623 machine catches fire. Another quite important intent is to allow people
624 to fiddle around with optimization.
626 By default, the only documentation that will be built is this tutorial
627 in @file{.info} format. To build the GiNaC tutorial and reference manual
628 in HTML, DVI, PostScript, or PDF formats, use one of
637 Generally, the top-level Makefile runs recursively to the
638 subdirectories. It is therefore safe to go into any subdirectory
639 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
640 @var{target} there in case something went wrong.
643 @node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
644 @c node-name, next, previous, up
645 @section Installing GiNaC
648 To install GiNaC on your system, simply type
654 As described in the section about configuration the files will be
655 installed in the following directories (the directories will be created
656 if they don't already exist):
661 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
662 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
663 So will @file{libginac.so} unless the configure script was
664 given the option @option{--disable-shared}. The proper symlinks
665 will be established as well.
668 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
669 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
672 All documentation (info) will be stuffed into
673 @file{@var{PREFIX}/share/doc/GiNaC/} (or
674 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
678 For the sake of completeness we will list some other useful make
679 targets: @command{make clean} deletes all files generated by
680 @command{make}, i.e. all the object files. In addition @command{make
681 distclean} removes all files generated by the configuration and
682 @command{make maintainer-clean} goes one step further and deletes files
683 that may require special tools to rebuild (like the @command{libtool}
684 for instance). Finally @command{make uninstall} removes the installed
685 library, header files and documentation@footnote{Uninstallation does not
686 work after you have called @command{make distclean} since the
687 @file{Makefile} is itself generated by the configuration from
688 @file{Makefile.in} and hence deleted by @command{make distclean}. There
689 are two obvious ways out of this dilemma. First, you can run the
690 configuration again with the same @var{PREFIX} thus creating a
691 @file{Makefile} with a working @samp{uninstall} target. Second, you can
692 do it by hand since you now know where all the files went during
696 @node Basic Concepts, Expressions, Installing GiNaC, Top
697 @c node-name, next, previous, up
698 @chapter Basic Concepts
700 This chapter will describe the different fundamental objects that can be
701 handled by GiNaC. But before doing so, it is worthwhile introducing you
702 to the more commonly used class of expressions, representing a flexible
703 meta-class for storing all mathematical objects.
706 * Expressions:: The fundamental GiNaC class.
707 * Automatic evaluation:: Evaluation and canonicalization.
708 * Error handling:: How the library reports errors.
709 * The Class Hierarchy:: Overview of GiNaC's classes.
710 * Symbols:: Symbolic objects.
711 * Numbers:: Numerical objects.
712 * Constants:: Pre-defined constants.
713 * Fundamental containers:: Sums, products and powers.
714 * Lists:: Lists of expressions.
715 * Mathematical functions:: Mathematical functions.
716 * Relations:: Equality, Inequality and all that.
717 * Integrals:: Symbolic integrals.
718 * Matrices:: Matrices.
719 * Indexed objects:: Handling indexed quantities.
720 * Non-commutative objects:: Algebras with non-commutative products.
721 * Hash Maps:: A faster alternative to std::map<>.
725 @node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
726 @c node-name, next, previous, up
728 @cindex expression (class @code{ex})
731 The most common class of objects a user deals with is the expression
732 @code{ex}, representing a mathematical object like a variable, number,
733 function, sum, product, etc@dots{} Expressions may be put together to form
734 new expressions, passed as arguments to functions, and so on. Here is a
735 little collection of valid expressions:
738 ex MyEx1 = 5; // simple number
739 ex MyEx2 = x + 2*y; // polynomial in x and y
740 ex MyEx3 = (x + 1)/(x - 1); // rational expression
741 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
742 ex MyEx5 = MyEx4 + 1; // similar to above
745 Expressions are handles to other more fundamental objects, that often
746 contain other expressions thus creating a tree of expressions
747 (@xref{Internal Structures}, for particular examples). Most methods on
748 @code{ex} therefore run top-down through such an expression tree. For
749 example, the method @code{has()} scans recursively for occurrences of
750 something inside an expression. Thus, if you have declared @code{MyEx4}
751 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
752 the argument of @code{sin} and hence return @code{true}.
754 The next sections will outline the general picture of GiNaC's class
755 hierarchy and describe the classes of objects that are handled by
758 @subsection Note: Expressions and STL containers
760 GiNaC expressions (@code{ex} objects) have value semantics (they can be
761 assigned, reassigned and copied like integral types) but the operator
762 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
763 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
765 This implies that in order to use expressions in sorted containers such as
766 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
767 comparison predicate. GiNaC provides such a predicate, called
768 @code{ex_is_less}. For example, a set of expressions should be defined
769 as @code{std::set<ex, ex_is_less>}.
771 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
772 don't pose a problem. A @code{std::vector<ex>} works as expected.
774 @xref{Information About Expressions}, for more about comparing and ordering
778 @node Automatic evaluation, Error handling, Expressions, Basic Concepts
779 @c node-name, next, previous, up
780 @section Automatic evaluation and canonicalization of expressions
783 GiNaC performs some automatic transformations on expressions, to simplify
784 them and put them into a canonical form. Some examples:
787 ex MyEx1 = 2*x - 1 + x; // 3*x-1
788 ex MyEx2 = x - x; // 0
789 ex MyEx3 = cos(2*Pi); // 1
790 ex MyEx4 = x*y/x; // y
793 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
794 evaluation}. GiNaC only performs transformations that are
798 at most of complexity
806 algebraically correct, possibly except for a set of measure zero (e.g.
807 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
810 There are two types of automatic transformations in GiNaC that may not
811 behave in an entirely obvious way at first glance:
815 The terms of sums and products (and some other things like the arguments of
816 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
817 into a canonical form that is deterministic, but not lexicographical or in
818 any other way easy to guess (it almost always depends on the number and
819 order of the symbols you define). However, constructing the same expression
820 twice, either implicitly or explicitly, will always result in the same
823 Expressions of the form 'number times sum' are automatically expanded (this
824 has to do with GiNaC's internal representation of sums and products). For
827 ex MyEx5 = 2*(x + y); // 2*x+2*y
828 ex MyEx6 = z*(x + y); // z*(x+y)
832 The general rule is that when you construct expressions, GiNaC automatically
833 creates them in canonical form, which might differ from the form you typed in
834 your program. This may create some awkward looking output (@samp{-y+x} instead
835 of @samp{x-y}) but allows for more efficient operation and usually yields
836 some immediate simplifications.
838 @cindex @code{eval()}
839 Internally, the anonymous evaluator in GiNaC is implemented by the methods
842 ex ex::eval(int level = 0) const;
843 ex basic::eval(int level = 0) const;
846 but unless you are extending GiNaC with your own classes or functions, there
847 should never be any reason to call them explicitly. All GiNaC methods that
848 transform expressions, like @code{subs()} or @code{normal()}, automatically
849 re-evaluate their results.
852 @node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
853 @c node-name, next, previous, up
854 @section Error handling
856 @cindex @code{pole_error} (class)
858 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
859 generated by GiNaC are subclassed from the standard @code{exception} class
860 defined in the @file{<stdexcept>} header. In addition to the predefined
861 @code{logic_error}, @code{domain_error}, @code{out_of_range},
862 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
863 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
864 exception that gets thrown when trying to evaluate a mathematical function
867 The @code{pole_error} class has a member function
870 int pole_error::degree() const;
873 that returns the order of the singularity (or 0 when the pole is
874 logarithmic or the order is undefined).
876 When using GiNaC it is useful to arrange for exceptions to be caught in
877 the main program even if you don't want to do any special error handling.
878 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
879 default exception handler of your C++ compiler's run-time system which
880 usually only aborts the program without giving any information what went
883 Here is an example for a @code{main()} function that catches and prints
884 exceptions generated by GiNaC:
889 #include <ginac/ginac.h>
891 using namespace GiNaC;
899 @} catch (exception &p) @{
900 cerr << p.what() << endl;
908 @node The Class Hierarchy, Symbols, Error handling, Basic Concepts
909 @c node-name, next, previous, up
910 @section The Class Hierarchy
912 GiNaC's class hierarchy consists of several classes representing
913 mathematical objects, all of which (except for @code{ex} and some
914 helpers) are internally derived from one abstract base class called
915 @code{basic}. You do not have to deal with objects of class
916 @code{basic}, instead you'll be dealing with symbols, numbers,
917 containers of expressions and so on.
921 To get an idea about what kinds of symbolic composites may be built we
922 have a look at the most important classes in the class hierarchy and
923 some of the relations among the classes:
925 @image{classhierarchy}
927 The abstract classes shown here (the ones without drop-shadow) are of no
928 interest for the user. They are used internally in order to avoid code
929 duplication if two or more classes derived from them share certain
930 features. An example is @code{expairseq}, a container for a sequence of
931 pairs each consisting of one expression and a number (@code{numeric}).
932 What @emph{is} visible to the user are the derived classes @code{add}
933 and @code{mul}, representing sums and products. @xref{Internal
934 Structures}, where these two classes are described in more detail. The
935 following table shortly summarizes what kinds of mathematical objects
936 are stored in the different classes:
939 @multitable @columnfractions .22 .78
940 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
941 @item @code{constant} @tab Constants like
948 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
949 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
950 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
951 @item @code{ncmul} @tab Products of non-commutative objects
952 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
957 @code{sqrt(}@math{2}@code{)}
960 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
961 @item @code{function} @tab A symbolic function like
968 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
969 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
970 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
971 @item @code{indexed} @tab Indexed object like @math{A_ij}
972 @item @code{tensor} @tab Special tensor like the delta and metric tensors
973 @item @code{idx} @tab Index of an indexed object
974 @item @code{varidx} @tab Index with variance
975 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
976 @item @code{wildcard} @tab Wildcard for pattern matching
977 @item @code{structure} @tab Template for user-defined classes
982 @node Symbols, Numbers, The Class Hierarchy, Basic Concepts
983 @c node-name, next, previous, up
985 @cindex @code{symbol} (class)
986 @cindex hierarchy of classes
989 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
990 manipulation what atoms are for chemistry.
992 A typical symbol definition looks like this:
997 This definition actually contains three very different things:
999 @item a C++ variable named @code{x}
1000 @item a @code{symbol} object stored in this C++ variable; this object
1001 represents the symbol in a GiNaC expression
1002 @item the string @code{"x"} which is the name of the symbol, used (almost)
1003 exclusively for printing expressions holding the symbol
1006 Symbols have an explicit name, supplied as a string during construction,
1007 because in C++, variable names can't be used as values, and the C++ compiler
1008 throws them away during compilation.
1010 It is possible to omit the symbol name in the definition:
1015 In this case, GiNaC will assign the symbol an internal, unique name of the
1016 form @code{symbolNNN}. This won't affect the usability of the symbol but
1017 the output of your calculations will become more readable if you give your
1018 symbols sensible names (for intermediate expressions that are only used
1019 internally such anonymous symbols can be quite useful, however).
1021 Now, here is one important property of GiNaC that differentiates it from
1022 other computer algebra programs you may have used: GiNaC does @emph{not} use
1023 the names of symbols to tell them apart, but a (hidden) serial number that
1024 is unique for each newly created @code{symbol} object. In you want to use
1025 one and the same symbol in different places in your program, you must only
1026 create one @code{symbol} object and pass that around. If you create another
1027 symbol, even if it has the same name, GiNaC will treat it as a different
1044 // prints "x^6" which looks right, but...
1046 cout << e.degree(x) << endl;
1047 // ...this doesn't work. The symbol "x" here is different from the one
1048 // in f() and in the expression returned by f(). Consequently, it
1053 One possibility to ensure that @code{f()} and @code{main()} use the same
1054 symbol is to pass the symbol as an argument to @code{f()}:
1056 ex f(int n, const ex & x)
1065 // Now, f() uses the same symbol.
1068 cout << e.degree(x) << endl;
1069 // prints "6", as expected
1073 Another possibility would be to define a global symbol @code{x} that is used
1074 by both @code{f()} and @code{main()}. If you are using global symbols and
1075 multiple compilation units you must take special care, however. Suppose
1076 that you have a header file @file{globals.h} in your program that defines
1077 a @code{symbol x("x");}. In this case, every unit that includes
1078 @file{globals.h} would also get its own definition of @code{x} (because
1079 header files are just inlined into the source code by the C++ preprocessor),
1080 and hence you would again end up with multiple equally-named, but different,
1081 symbols. Instead, the @file{globals.h} header should only contain a
1082 @emph{declaration} like @code{extern symbol x;}, with the definition of
1083 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1085 A different approach to ensuring that symbols used in different parts of
1086 your program are identical is to create them with a @emph{factory} function
1089 const symbol & get_symbol(const string & s)
1091 static map<string, symbol> directory;
1092 map<string, symbol>::iterator i = directory.find(s);
1093 if (i != directory.end())
1096 return directory.insert(make_pair(s, symbol(s))).first->second;
1100 This function returns one newly constructed symbol for each name that is
1101 passed in, and it returns the same symbol when called multiple times with
1102 the same name. Using this symbol factory, we can rewrite our example like
1107 return pow(get_symbol("x"), n);
1114 // Both calls of get_symbol("x") yield the same symbol.
1115 cout << e.degree(get_symbol("x")) << endl;
1120 Instead of creating symbols from strings we could also have
1121 @code{get_symbol()} take, for example, an integer number as its argument.
1122 In this case, we would probably want to give the generated symbols names
1123 that include this number, which can be accomplished with the help of an
1124 @code{ostringstream}.
1126 In general, if you're getting weird results from GiNaC such as an expression
1127 @samp{x-x} that is not simplified to zero, you should check your symbol
1130 As we said, the names of symbols primarily serve for purposes of expression
1131 output. But there are actually two instances where GiNaC uses the names for
1132 identifying symbols: When constructing an expression from a string, and when
1133 recreating an expression from an archive (@pxref{Input/Output}).
1135 In addition to its name, a symbol may contain a special string that is used
1138 symbol x("x", "\\Box");
1141 This creates a symbol that is printed as "@code{x}" in normal output, but
1142 as "@code{\Box}" in LaTeX code (@xref{Input/Output}, for more
1143 information about the different output formats of expressions in GiNaC).
1144 GiNaC automatically creates proper LaTeX code for symbols having names of
1145 greek letters (@samp{alpha}, @samp{mu}, etc.).
1147 @cindex @code{subs()}
1148 Symbols in GiNaC can't be assigned values. If you need to store results of
1149 calculations and give them a name, use C++ variables of type @code{ex}.
1150 If you want to replace a symbol in an expression with something else, you
1151 can invoke the expression's @code{.subs()} method
1152 (@pxref{Substituting Expressions}).
1154 @cindex @code{realsymbol()}
1155 By default, symbols are expected to stand in for complex values, i.e. they live
1156 in the complex domain. As a consequence, operations like complex conjugation,
1157 for example (@pxref{Complex Conjugation}), do @emph{not} evaluate if applied
1158 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1159 because of the unknown imaginary part of @code{x}.
1160 On the other hand, if you are sure that your symbols will hold only real values, you
1161 would like to have such functions evaluated. Therefore GiNaC allows you to specify
1162 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1163 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1166 @node Numbers, Constants, Symbols, Basic Concepts
1167 @c node-name, next, previous, up
1169 @cindex @code{numeric} (class)
1175 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1176 The classes therein serve as foundation classes for GiNaC. CLN stands
1177 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1178 In order to find out more about CLN's internals, the reader is referred to
1179 the documentation of that library. @inforef{Introduction, , cln}, for
1180 more information. Suffice to say that it is by itself build on top of
1181 another library, the GNU Multiple Precision library GMP, which is an
1182 extremely fast library for arbitrary long integers and rationals as well
1183 as arbitrary precision floating point numbers. It is very commonly used
1184 by several popular cryptographic applications. CLN extends GMP by
1185 several useful things: First, it introduces the complex number field
1186 over either reals (i.e. floating point numbers with arbitrary precision)
1187 or rationals. Second, it automatically converts rationals to integers
1188 if the denominator is unity and complex numbers to real numbers if the
1189 imaginary part vanishes and also correctly treats algebraic functions.
1190 Third it provides good implementations of state-of-the-art algorithms
1191 for all trigonometric and hyperbolic functions as well as for
1192 calculation of some useful constants.
1194 The user can construct an object of class @code{numeric} in several
1195 ways. The following example shows the four most important constructors.
1196 It uses construction from C-integer, construction of fractions from two
1197 integers, construction from C-float and construction from a string:
1201 #include <ginac/ginac.h>
1202 using namespace GiNaC;
1206 numeric two = 2; // exact integer 2
1207 numeric r(2,3); // exact fraction 2/3
1208 numeric e(2.71828); // floating point number
1209 numeric p = "3.14159265358979323846"; // constructor from string
1210 // Trott's constant in scientific notation:
1211 numeric trott("1.0841015122311136151E-2");
1213 std::cout << two*p << std::endl; // floating point 6.283...
1218 @cindex complex numbers
1219 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1224 numeric z1 = 2-3*I; // exact complex number 2-3i
1225 numeric z2 = 5.9+1.6*I; // complex floating point number
1229 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1230 This would, however, call C's built-in operator @code{/} for integers
1231 first and result in a numeric holding a plain integer 1. @strong{Never
1232 use the operator @code{/} on integers} unless you know exactly what you
1233 are doing! Use the constructor from two integers instead, as shown in
1234 the example above. Writing @code{numeric(1)/2} may look funny but works
1237 @cindex @code{Digits}
1239 We have seen now the distinction between exact numbers and floating
1240 point numbers. Clearly, the user should never have to worry about
1241 dynamically created exact numbers, since their `exactness' always
1242 determines how they ought to be handled, i.e. how `long' they are. The
1243 situation is different for floating point numbers. Their accuracy is
1244 controlled by one @emph{global} variable, called @code{Digits}. (For
1245 those readers who know about Maple: it behaves very much like Maple's
1246 @code{Digits}). All objects of class numeric that are constructed from
1247 then on will be stored with a precision matching that number of decimal
1252 #include <ginac/ginac.h>
1253 using namespace std;
1254 using namespace GiNaC;
1258 numeric three(3.0), one(1.0);
1259 numeric x = one/three;
1261 cout << "in " << Digits << " digits:" << endl;
1263 cout << Pi.evalf() << endl;
1275 The above example prints the following output to screen:
1279 0.33333333333333333334
1280 3.1415926535897932385
1282 0.33333333333333333333333333333333333333333333333333333333333333333334
1283 3.1415926535897932384626433832795028841971693993751058209749445923078
1287 Note that the last number is not necessarily rounded as you would
1288 naively expect it to be rounded in the decimal system. But note also,
1289 that in both cases you got a couple of extra digits. This is because
1290 numbers are internally stored by CLN as chunks of binary digits in order
1291 to match your machine's word size and to not waste precision. Thus, on
1292 architectures with different word size, the above output might even
1293 differ with regard to actually computed digits.
1295 It should be clear that objects of class @code{numeric} should be used
1296 for constructing numbers or for doing arithmetic with them. The objects
1297 one deals with most of the time are the polymorphic expressions @code{ex}.
1299 @subsection Tests on numbers
1301 Once you have declared some numbers, assigned them to expressions and
1302 done some arithmetic with them it is frequently desired to retrieve some
1303 kind of information from them like asking whether that number is
1304 integer, rational, real or complex. For those cases GiNaC provides
1305 several useful methods. (Internally, they fall back to invocations of
1306 certain CLN functions.)
1308 As an example, let's construct some rational number, multiply it with
1309 some multiple of its denominator and test what comes out:
1313 #include <ginac/ginac.h>
1314 using namespace std;
1315 using namespace GiNaC;
1317 // some very important constants:
1318 const numeric twentyone(21);
1319 const numeric ten(10);
1320 const numeric five(5);
1324 numeric answer = twentyone;
1327 cout << answer.is_integer() << endl; // false, it's 21/5
1329 cout << answer.is_integer() << endl; // true, it's 42 now!
1333 Note that the variable @code{answer} is constructed here as an integer
1334 by @code{numeric}'s copy constructor but in an intermediate step it
1335 holds a rational number represented as integer numerator and integer
1336 denominator. When multiplied by 10, the denominator becomes unity and
1337 the result is automatically converted to a pure integer again.
1338 Internally, the underlying CLN is responsible for this behavior and we
1339 refer the reader to CLN's documentation. Suffice to say that
1340 the same behavior applies to complex numbers as well as return values of
1341 certain functions. Complex numbers are automatically converted to real
1342 numbers if the imaginary part becomes zero. The full set of tests that
1343 can be applied is listed in the following table.
1346 @multitable @columnfractions .30 .70
1347 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1348 @item @code{.is_zero()}
1349 @tab @dots{}equal to zero
1350 @item @code{.is_positive()}
1351 @tab @dots{}not complex and greater than 0
1352 @item @code{.is_integer()}
1353 @tab @dots{}a (non-complex) integer
1354 @item @code{.is_pos_integer()}
1355 @tab @dots{}an integer and greater than 0
1356 @item @code{.is_nonneg_integer()}
1357 @tab @dots{}an integer and greater equal 0
1358 @item @code{.is_even()}
1359 @tab @dots{}an even integer
1360 @item @code{.is_odd()}
1361 @tab @dots{}an odd integer
1362 @item @code{.is_prime()}
1363 @tab @dots{}a prime integer (probabilistic primality test)
1364 @item @code{.is_rational()}
1365 @tab @dots{}an exact rational number (integers are rational, too)
1366 @item @code{.is_real()}
1367 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1368 @item @code{.is_cinteger()}
1369 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1370 @item @code{.is_crational()}
1371 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1375 @subsection Numeric functions
1377 The following functions can be applied to @code{numeric} objects and will be
1378 evaluated immediately:
1381 @multitable @columnfractions .30 .70
1382 @item @strong{Name} @tab @strong{Function}
1383 @item @code{inverse(z)}
1384 @tab returns @math{1/z}
1385 @cindex @code{inverse()} (numeric)
1386 @item @code{pow(a, b)}
1387 @tab exponentiation @math{a^b}
1390 @item @code{real(z)}
1392 @cindex @code{real()}
1393 @item @code{imag(z)}
1395 @cindex @code{imag()}
1396 @item @code{csgn(z)}
1397 @tab complex sign (returns an @code{int})
1398 @item @code{numer(z)}
1399 @tab numerator of rational or complex rational number
1400 @item @code{denom(z)}
1401 @tab denominator of rational or complex rational number
1402 @item @code{sqrt(z)}
1404 @item @code{isqrt(n)}
1405 @tab integer square root
1406 @cindex @code{isqrt()}
1413 @item @code{asin(z)}
1415 @item @code{acos(z)}
1417 @item @code{atan(z)}
1418 @tab inverse tangent
1419 @item @code{atan(y, x)}
1420 @tab inverse tangent with two arguments
1421 @item @code{sinh(z)}
1422 @tab hyperbolic sine
1423 @item @code{cosh(z)}
1424 @tab hyperbolic cosine
1425 @item @code{tanh(z)}
1426 @tab hyperbolic tangent
1427 @item @code{asinh(z)}
1428 @tab inverse hyperbolic sine
1429 @item @code{acosh(z)}
1430 @tab inverse hyperbolic cosine
1431 @item @code{atanh(z)}
1432 @tab inverse hyperbolic tangent
1434 @tab exponential function
1436 @tab natural logarithm
1439 @item @code{zeta(z)}
1440 @tab Riemann's zeta function
1441 @item @code{tgamma(z)}
1443 @item @code{lgamma(z)}
1444 @tab logarithm of gamma function
1446 @tab psi (digamma) function
1447 @item @code{psi(n, z)}
1448 @tab derivatives of psi function (polygamma functions)
1449 @item @code{factorial(n)}
1450 @tab factorial function @math{n!}
1451 @item @code{doublefactorial(n)}
1452 @tab double factorial function @math{n!!}
1453 @cindex @code{doublefactorial()}
1454 @item @code{binomial(n, k)}
1455 @tab binomial coefficients
1456 @item @code{bernoulli(n)}
1457 @tab Bernoulli numbers
1458 @cindex @code{bernoulli()}
1459 @item @code{fibonacci(n)}
1460 @tab Fibonacci numbers
1461 @cindex @code{fibonacci()}
1462 @item @code{mod(a, b)}
1463 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1464 @cindex @code{mod()}
1465 @item @code{smod(a, b)}
1466 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1467 @cindex @code{smod()}
1468 @item @code{irem(a, b)}
1469 @tab integer remainder (has the sign of @math{a}, or is zero)
1470 @cindex @code{irem()}
1471 @item @code{irem(a, b, q)}
1472 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1473 @item @code{iquo(a, b)}
1474 @tab integer quotient
1475 @cindex @code{iquo()}
1476 @item @code{iquo(a, b, r)}
1477 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1478 @item @code{gcd(a, b)}
1479 @tab greatest common divisor
1480 @item @code{lcm(a, b)}
1481 @tab least common multiple
1485 Most of these functions are also available as symbolic functions that can be
1486 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1487 as polynomial algorithms.
1489 @subsection Converting numbers
1491 Sometimes it is desirable to convert a @code{numeric} object back to a
1492 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1493 class provides a couple of methods for this purpose:
1495 @cindex @code{to_int()}
1496 @cindex @code{to_long()}
1497 @cindex @code{to_double()}
1498 @cindex @code{to_cl_N()}
1500 int numeric::to_int() const;
1501 long numeric::to_long() const;
1502 double numeric::to_double() const;
1503 cln::cl_N numeric::to_cl_N() const;
1506 @code{to_int()} and @code{to_long()} only work when the number they are
1507 applied on is an exact integer. Otherwise the program will halt with a
1508 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1509 rational number will return a floating-point approximation. Both
1510 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1511 part of complex numbers.
1514 @node Constants, Fundamental containers, Numbers, Basic Concepts
1515 @c node-name, next, previous, up
1517 @cindex @code{constant} (class)
1520 @cindex @code{Catalan}
1521 @cindex @code{Euler}
1522 @cindex @code{evalf()}
1523 Constants behave pretty much like symbols except that they return some
1524 specific number when the method @code{.evalf()} is called.
1526 The predefined known constants are:
1529 @multitable @columnfractions .14 .30 .56
1530 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1532 @tab Archimedes' constant
1533 @tab 3.14159265358979323846264338327950288
1534 @item @code{Catalan}
1535 @tab Catalan's constant
1536 @tab 0.91596559417721901505460351493238411
1538 @tab Euler's (or Euler-Mascheroni) constant
1539 @tab 0.57721566490153286060651209008240243
1544 @node Fundamental containers, Lists, Constants, Basic Concepts
1545 @c node-name, next, previous, up
1546 @section Sums, products and powers
1550 @cindex @code{power}
1552 Simple rational expressions are written down in GiNaC pretty much like
1553 in other CAS or like expressions involving numerical variables in C.
1554 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1555 been overloaded to achieve this goal. When you run the following
1556 code snippet, the constructor for an object of type @code{mul} is
1557 automatically called to hold the product of @code{a} and @code{b} and
1558 then the constructor for an object of type @code{add} is called to hold
1559 the sum of that @code{mul} object and the number one:
1563 symbol a("a"), b("b");
1568 @cindex @code{pow()}
1569 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1570 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1571 construction is necessary since we cannot safely overload the constructor
1572 @code{^} in C++ to construct a @code{power} object. If we did, it would
1573 have several counterintuitive and undesired effects:
1577 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1579 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1580 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1581 interpret this as @code{x^(a^b)}.
1583 Also, expressions involving integer exponents are very frequently used,
1584 which makes it even more dangerous to overload @code{^} since it is then
1585 hard to distinguish between the semantics as exponentiation and the one
1586 for exclusive or. (It would be embarrassing to return @code{1} where one
1587 has requested @code{2^3}.)
1590 @cindex @command{ginsh}
1591 All effects are contrary to mathematical notation and differ from the
1592 way most other CAS handle exponentiation, therefore overloading @code{^}
1593 is ruled out for GiNaC's C++ part. The situation is different in
1594 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1595 that the other frequently used exponentiation operator @code{**} does
1596 not exist at all in C++).
1598 To be somewhat more precise, objects of the three classes described
1599 here, are all containers for other expressions. An object of class
1600 @code{power} is best viewed as a container with two slots, one for the
1601 basis, one for the exponent. All valid GiNaC expressions can be
1602 inserted. However, basic transformations like simplifying
1603 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1604 when this is mathematically possible. If we replace the outer exponent
1605 three in the example by some symbols @code{a}, the simplification is not
1606 safe and will not be performed, since @code{a} might be @code{1/2} and
1609 Objects of type @code{add} and @code{mul} are containers with an
1610 arbitrary number of slots for expressions to be inserted. Again, simple
1611 and safe simplifications are carried out like transforming
1612 @code{3*x+4-x} to @code{2*x+4}.
1615 @node Lists, Mathematical functions, Fundamental containers, Basic Concepts
1616 @c node-name, next, previous, up
1617 @section Lists of expressions
1618 @cindex @code{lst} (class)
1620 @cindex @code{nops()}
1622 @cindex @code{append()}
1623 @cindex @code{prepend()}
1624 @cindex @code{remove_first()}
1625 @cindex @code{remove_last()}
1626 @cindex @code{remove_all()}
1628 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1629 expressions. They are not as ubiquitous as in many other computer algebra
1630 packages, but are sometimes used to supply a variable number of arguments of
1631 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1632 constructors, so you should have a basic understanding of them.
1634 Lists can be constructed by assigning a comma-separated sequence of
1639 symbol x("x"), y("y");
1642 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1647 There are also constructors that allow direct creation of lists of up to
1648 16 expressions, which is often more convenient but slightly less efficient:
1652 // This produces the same list 'l' as above:
1653 // lst l(x, 2, y, x+y);
1654 // lst l = lst(x, 2, y, x+y);
1658 Use the @code{nops()} method to determine the size (number of expressions) of
1659 a list and the @code{op()} method or the @code{[]} operator to access
1660 individual elements:
1664 cout << l.nops() << endl; // prints '4'
1665 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1669 As with the standard @code{list<T>} container, accessing random elements of a
1670 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1671 sequential access to the elements of a list is possible with the
1672 iterator types provided by the @code{lst} class:
1675 typedef ... lst::const_iterator;
1676 typedef ... lst::const_reverse_iterator;
1677 lst::const_iterator lst::begin() const;
1678 lst::const_iterator lst::end() const;
1679 lst::const_reverse_iterator lst::rbegin() const;
1680 lst::const_reverse_iterator lst::rend() const;
1683 For example, to print the elements of a list individually you can use:
1688 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1693 which is one order faster than
1698 for (size_t i = 0; i < l.nops(); ++i)
1699 cout << l.op(i) << endl;
1703 These iterators also allow you to use some of the algorithms provided by
1704 the C++ standard library:
1708 // print the elements of the list (requires #include <iterator>)
1709 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1711 // sum up the elements of the list (requires #include <numeric>)
1712 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1713 cout << sum << endl; // prints '2+2*x+2*y'
1717 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1718 (the only other one is @code{matrix}). You can modify single elements:
1722 l[1] = 42; // l is now @{x, 42, y, x+y@}
1723 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1727 You can append or prepend an expression to a list with the @code{append()}
1728 and @code{prepend()} methods:
1732 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1733 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1737 You can remove the first or last element of a list with @code{remove_first()}
1738 and @code{remove_last()}:
1742 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1743 l.remove_last(); // l is now @{x, 7, y, x+y@}
1747 You can remove all the elements of a list with @code{remove_all()}:
1751 l.remove_all(); // l is now empty
1755 You can bring the elements of a list into a canonical order with @code{sort()}:
1764 // l1 and l2 are now equal
1768 Finally, you can remove all but the first element of consecutive groups of
1769 elements with @code{unique()}:
1774 l3 = x, 2, 2, 2, y, x+y, y+x;
1775 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1780 @node Mathematical functions, Relations, Lists, Basic Concepts
1781 @c node-name, next, previous, up
1782 @section Mathematical functions
1783 @cindex @code{function} (class)
1784 @cindex trigonometric function
1785 @cindex hyperbolic function
1787 There are quite a number of useful functions hard-wired into GiNaC. For
1788 instance, all trigonometric and hyperbolic functions are implemented
1789 (@xref{Built-in Functions}, for a complete list).
1791 These functions (better called @emph{pseudofunctions}) are all objects
1792 of class @code{function}. They accept one or more expressions as
1793 arguments and return one expression. If the arguments are not
1794 numerical, the evaluation of the function may be halted, as it does in
1795 the next example, showing how a function returns itself twice and
1796 finally an expression that may be really useful:
1798 @cindex Gamma function
1799 @cindex @code{subs()}
1802 symbol x("x"), y("y");
1804 cout << tgamma(foo) << endl;
1805 // -> tgamma(x+(1/2)*y)
1806 ex bar = foo.subs(y==1);
1807 cout << tgamma(bar) << endl;
1809 ex foobar = bar.subs(x==7);
1810 cout << tgamma(foobar) << endl;
1811 // -> (135135/128)*Pi^(1/2)
1815 Besides evaluation most of these functions allow differentiation, series
1816 expansion and so on. Read the next chapter in order to learn more about
1819 It must be noted that these pseudofunctions are created by inline
1820 functions, where the argument list is templated. This means that
1821 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1822 @code{sin(ex(1))} and will therefore not result in a floating point
1823 number. Unless of course the function prototype is explicitly
1824 overridden -- which is the case for arguments of type @code{numeric}
1825 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1826 point number of class @code{numeric} you should call
1827 @code{sin(numeric(1))}. This is almost the same as calling
1828 @code{sin(1).evalf()} except that the latter will return a numeric
1829 wrapped inside an @code{ex}.
1832 @node Relations, Integrals, Mathematical functions, Basic Concepts
1833 @c node-name, next, previous, up
1835 @cindex @code{relational} (class)
1837 Sometimes, a relation holding between two expressions must be stored
1838 somehow. The class @code{relational} is a convenient container for such
1839 purposes. A relation is by definition a container for two @code{ex} and
1840 a relation between them that signals equality, inequality and so on.
1841 They are created by simply using the C++ operators @code{==}, @code{!=},
1842 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1844 @xref{Mathematical functions}, for examples where various applications
1845 of the @code{.subs()} method show how objects of class relational are
1846 used as arguments. There they provide an intuitive syntax for
1847 substitutions. They are also used as arguments to the @code{ex::series}
1848 method, where the left hand side of the relation specifies the variable
1849 to expand in and the right hand side the expansion point. They can also
1850 be used for creating systems of equations that are to be solved for
1851 unknown variables. But the most common usage of objects of this class
1852 is rather inconspicuous in statements of the form @code{if
1853 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1854 conversion from @code{relational} to @code{bool} takes place. Note,
1855 however, that @code{==} here does not perform any simplifications, hence
1856 @code{expand()} must be called explicitly.
1858 @node Integrals, Matrices, Relations, Basic Concepts
1859 @c node-name, next, previous, up
1861 @cindex @code{integral} (class)
1863 An object of class @dfn{integral} can be used to hold a symbolic integral.
1864 If you want to symbolically represent the integral of @code{x*x} from 0 to
1865 1, you would write this as
1867 integral(x, 0, 1, x*x)
1869 The first argument is the integration variable. It should be noted that
1870 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1871 fact, it can only integrate polynomials. An expression containing integrals
1872 can be evaluated symbolically by calling the
1876 method on it. Numerical evaluation is available by calling the
1880 method on an expression containing the integral. This will only evaluate
1881 integrals into a number if @code{subs}ing the integration variable by a
1882 number in the fourth argument of an integral and then @code{evalf}ing the
1883 result always results in a number. Of course, also the boundaries of the
1884 integration domain must @code{evalf} into numbers. It should be noted that
1885 trying to @code{evalf} a function with discontinuities in the integration
1886 domain is not recommended. The accuracy of the numeric evaluation of
1887 integrals is determined by the static member variable
1889 ex integral::relative_integration_error
1891 of the class @code{integral}. The default value of this is 10^-8.
1892 The integration works by halving the interval of integration, until numeric
1893 stability of the answer indicates that the requested accuracy has been
1894 reached. The maximum depth of the halving can be set via the static member
1897 int integral::max_integration_level
1899 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1900 return the integral unevaluated. The function that performs the numerical
1901 evaluation, is also available as
1903 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1906 This function will throw an exception if the maximum depth is exceeded. The
1907 last parameter of the function is optional and defaults to the
1908 @code{relative_integration_error}. To make sure that we do not do too
1909 much work if an expression contains the same integral multiple times,
1910 a lookup table is used.
1912 If you know that an expression holds an integral, you can get the
1913 integration variable, the left boundary, right boundary and integrand by
1914 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1915 @code{.op(3)}. Differentiating integrals with respect to variables works
1916 as expected. Note that it makes no sense to differentiate an integral
1917 with respect to the integration variable.
1919 @node Matrices, Indexed objects, Integrals, Basic Concepts
1920 @c node-name, next, previous, up
1922 @cindex @code{matrix} (class)
1924 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1925 matrix with @math{m} rows and @math{n} columns are accessed with two
1926 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1927 second one in the range 0@dots{}@math{n-1}.
1929 There are a couple of ways to construct matrices, with or without preset
1930 elements. The constructor
1933 matrix::matrix(unsigned r, unsigned c);
1936 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1939 The fastest way to create a matrix with preinitialized elements is to assign
1940 a list of comma-separated expressions to an empty matrix (see below for an
1941 example). But you can also specify the elements as a (flat) list with
1944 matrix::matrix(unsigned r, unsigned c, const lst & l);
1949 @cindex @code{lst_to_matrix()}
1951 ex lst_to_matrix(const lst & l);
1954 constructs a matrix from a list of lists, each list representing a matrix row.
1956 There is also a set of functions for creating some special types of
1959 @cindex @code{diag_matrix()}
1960 @cindex @code{unit_matrix()}
1961 @cindex @code{symbolic_matrix()}
1963 ex diag_matrix(const lst & l);
1964 ex unit_matrix(unsigned x);
1965 ex unit_matrix(unsigned r, unsigned c);
1966 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1967 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1968 const string & tex_base_name);
1971 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1972 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1973 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1974 matrix filled with newly generated symbols made of the specified base name
1975 and the position of each element in the matrix.
1977 Matrix elements can be accessed and set using the parenthesis (function call)
1981 const ex & matrix::operator()(unsigned r, unsigned c) const;
1982 ex & matrix::operator()(unsigned r, unsigned c);
1985 It is also possible to access the matrix elements in a linear fashion with
1986 the @code{op()} method. But C++-style subscripting with square brackets
1987 @samp{[]} is not available.
1989 Here are a couple of examples for constructing matrices:
1993 symbol a("a"), b("b");
2007 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2010 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2013 cout << diag_matrix(lst(a, b)) << endl;
2016 cout << unit_matrix(3) << endl;
2017 // -> [[1,0,0],[0,1,0],[0,0,1]]
2019 cout << symbolic_matrix(2, 3, "x") << endl;
2020 // -> [[x00,x01,x02],[x10,x11,x12]]
2024 @cindex @code{transpose()}
2025 There are three ways to do arithmetic with matrices. The first (and most
2026 direct one) is to use the methods provided by the @code{matrix} class:
2029 matrix matrix::add(const matrix & other) const;
2030 matrix matrix::sub(const matrix & other) const;
2031 matrix matrix::mul(const matrix & other) const;
2032 matrix matrix::mul_scalar(const ex & other) const;
2033 matrix matrix::pow(const ex & expn) const;
2034 matrix matrix::transpose() const;
2037 All of these methods return the result as a new matrix object. Here is an
2038 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2043 matrix A(2, 2), B(2, 2), C(2, 2);
2051 matrix result = A.mul(B).sub(C.mul_scalar(2));
2052 cout << result << endl;
2053 // -> [[-13,-6],[1,2]]
2058 @cindex @code{evalm()}
2059 The second (and probably the most natural) way is to construct an expression
2060 containing matrices with the usual arithmetic operators and @code{pow()}.
2061 For efficiency reasons, expressions with sums, products and powers of
2062 matrices are not automatically evaluated in GiNaC. You have to call the
2066 ex ex::evalm() const;
2069 to obtain the result:
2076 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2077 cout << e.evalm() << endl;
2078 // -> [[-13,-6],[1,2]]
2083 The non-commutativity of the product @code{A*B} in this example is
2084 automatically recognized by GiNaC. There is no need to use a special
2085 operator here. @xref{Non-commutative objects}, for more information about
2086 dealing with non-commutative expressions.
2088 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2089 to perform the arithmetic:
2094 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2095 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2097 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2098 cout << e.simplify_indexed() << endl;
2099 // -> [[-13,-6],[1,2]].i.j
2103 Using indices is most useful when working with rectangular matrices and
2104 one-dimensional vectors because you don't have to worry about having to
2105 transpose matrices before multiplying them. @xref{Indexed objects}, for
2106 more information about using matrices with indices, and about indices in
2109 The @code{matrix} class provides a couple of additional methods for
2110 computing determinants, traces, characteristic polynomials and ranks:
2112 @cindex @code{determinant()}
2113 @cindex @code{trace()}
2114 @cindex @code{charpoly()}
2115 @cindex @code{rank()}
2117 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2118 ex matrix::trace() const;
2119 ex matrix::charpoly(const ex & lambda) const;
2120 unsigned matrix::rank() const;
2123 The @samp{algo} argument of @code{determinant()} allows to select
2124 between different algorithms for calculating the determinant. The
2125 asymptotic speed (as parametrized by the matrix size) can greatly differ
2126 between those algorithms, depending on the nature of the matrix'
2127 entries. The possible values are defined in the @file{flags.h} header
2128 file. By default, GiNaC uses a heuristic to automatically select an
2129 algorithm that is likely (but not guaranteed) to give the result most
2132 @cindex @code{inverse()} (matrix)
2133 @cindex @code{solve()}
2134 Matrices may also be inverted using the @code{ex matrix::inverse()}
2135 method and linear systems may be solved with:
2138 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2139 unsigned algo=solve_algo::automatic) const;
2142 Assuming the matrix object this method is applied on is an @code{m}
2143 times @code{n} matrix, then @code{vars} must be a @code{n} times
2144 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2145 times @code{p} matrix. The returned matrix then has dimension @code{n}
2146 times @code{p} and in the case of an underdetermined system will still
2147 contain some of the indeterminates from @code{vars}. If the system is
2148 overdetermined, an exception is thrown.
2151 @node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
2152 @c node-name, next, previous, up
2153 @section Indexed objects
2155 GiNaC allows you to handle expressions containing general indexed objects in
2156 arbitrary spaces. It is also able to canonicalize and simplify such
2157 expressions and perform symbolic dummy index summations. There are a number
2158 of predefined indexed objects provided, like delta and metric tensors.
2160 There are few restrictions placed on indexed objects and their indices and
2161 it is easy to construct nonsense expressions, but our intention is to
2162 provide a general framework that allows you to implement algorithms with
2163 indexed quantities, getting in the way as little as possible.
2165 @cindex @code{idx} (class)
2166 @cindex @code{indexed} (class)
2167 @subsection Indexed quantities and their indices
2169 Indexed expressions in GiNaC are constructed of two special types of objects,
2170 @dfn{index objects} and @dfn{indexed objects}.
2174 @cindex contravariant
2177 @item Index objects are of class @code{idx} or a subclass. Every index has
2178 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2179 the index lives in) which can both be arbitrary expressions but are usually
2180 a number or a simple symbol. In addition, indices of class @code{varidx} have
2181 a @dfn{variance} (they can be co- or contravariant), and indices of class
2182 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2184 @item Indexed objects are of class @code{indexed} or a subclass. They
2185 contain a @dfn{base expression} (which is the expression being indexed), and
2186 one or more indices.
2190 @strong{Please notice:} when printing expressions, covariant indices and indices
2191 without variance are denoted @samp{.i} while contravariant indices are
2192 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2193 value. In the following, we are going to use that notation in the text so
2194 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2195 not visible in the output.
2197 A simple example shall illustrate the concepts:
2201 #include <ginac/ginac.h>
2202 using namespace std;
2203 using namespace GiNaC;
2207 symbol i_sym("i"), j_sym("j");
2208 idx i(i_sym, 3), j(j_sym, 3);
2211 cout << indexed(A, i, j) << endl;
2213 cout << index_dimensions << indexed(A, i, j) << endl;
2215 cout << dflt; // reset cout to default output format (dimensions hidden)
2219 The @code{idx} constructor takes two arguments, the index value and the
2220 index dimension. First we define two index objects, @code{i} and @code{j},
2221 both with the numeric dimension 3. The value of the index @code{i} is the
2222 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2223 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2224 construct an expression containing one indexed object, @samp{A.i.j}. It has
2225 the symbol @code{A} as its base expression and the two indices @code{i} and
2228 The dimensions of indices are normally not visible in the output, but one
2229 can request them to be printed with the @code{index_dimensions} manipulator,
2232 Note the difference between the indices @code{i} and @code{j} which are of
2233 class @code{idx}, and the index values which are the symbols @code{i_sym}
2234 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2235 or numbers but must be index objects. For example, the following is not
2236 correct and will raise an exception:
2239 symbol i("i"), j("j");
2240 e = indexed(A, i, j); // ERROR: indices must be of type idx
2243 You can have multiple indexed objects in an expression, index values can
2244 be numeric, and index dimensions symbolic:
2248 symbol B("B"), dim("dim");
2249 cout << 4 * indexed(A, i)
2250 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2255 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2256 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2257 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2258 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2259 @code{simplify_indexed()} for that, see below).
2261 In fact, base expressions, index values and index dimensions can be
2262 arbitrary expressions:
2266 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2271 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2272 get an error message from this but you will probably not be able to do
2273 anything useful with it.
2275 @cindex @code{get_value()}
2276 @cindex @code{get_dimension()}
2280 ex idx::get_value();
2281 ex idx::get_dimension();
2284 return the value and dimension of an @code{idx} object. If you have an index
2285 in an expression, such as returned by calling @code{.op()} on an indexed
2286 object, you can get a reference to the @code{idx} object with the function
2287 @code{ex_to<idx>()} on the expression.
2289 There are also the methods
2292 bool idx::is_numeric();
2293 bool idx::is_symbolic();
2294 bool idx::is_dim_numeric();
2295 bool idx::is_dim_symbolic();
2298 for checking whether the value and dimension are numeric or symbolic
2299 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2300 About Expressions}) returns information about the index value.
2302 @cindex @code{varidx} (class)
2303 If you need co- and contravariant indices, use the @code{varidx} class:
2307 symbol mu_sym("mu"), nu_sym("nu");
2308 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2309 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2311 cout << indexed(A, mu, nu) << endl;
2313 cout << indexed(A, mu_co, nu) << endl;
2315 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2320 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2321 co- or contravariant. The default is a contravariant (upper) index, but
2322 this can be overridden by supplying a third argument to the @code{varidx}
2323 constructor. The two methods
2326 bool varidx::is_covariant();
2327 bool varidx::is_contravariant();
2330 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2331 to get the object reference from an expression). There's also the very useful
2335 ex varidx::toggle_variance();
2338 which makes a new index with the same value and dimension but the opposite
2339 variance. By using it you only have to define the index once.
2341 @cindex @code{spinidx} (class)
2342 The @code{spinidx} class provides dotted and undotted variant indices, as
2343 used in the Weyl-van-der-Waerden spinor formalism:
2347 symbol K("K"), C_sym("C"), D_sym("D");
2348 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2349 // contravariant, undotted
2350 spinidx C_co(C_sym, 2, true); // covariant index
2351 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2352 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2354 cout << indexed(K, C, D) << endl;
2356 cout << indexed(K, C_co, D_dot) << endl;
2358 cout << indexed(K, D_co_dot, D) << endl;
2363 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2364 dotted or undotted. The default is undotted but this can be overridden by
2365 supplying a fourth argument to the @code{spinidx} constructor. The two
2369 bool spinidx::is_dotted();
2370 bool spinidx::is_undotted();
2373 allow you to check whether or not a @code{spinidx} object is dotted (use
2374 @code{ex_to<spinidx>()} to get the object reference from an expression).
2375 Finally, the two methods
2378 ex spinidx::toggle_dot();
2379 ex spinidx::toggle_variance_dot();
2382 create a new index with the same value and dimension but opposite dottedness
2383 and the same or opposite variance.
2385 @subsection Substituting indices
2387 @cindex @code{subs()}
2388 Sometimes you will want to substitute one symbolic index with another
2389 symbolic or numeric index, for example when calculating one specific element
2390 of a tensor expression. This is done with the @code{.subs()} method, as it
2391 is done for symbols (see @ref{Substituting Expressions}).
2393 You have two possibilities here. You can either substitute the whole index
2394 by another index or expression:
2398 ex e = indexed(A, mu_co);
2399 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2400 // -> A.mu becomes A~nu
2401 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2402 // -> A.mu becomes A~0
2403 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2404 // -> A.mu becomes A.0
2408 The third example shows that trying to replace an index with something that
2409 is not an index will substitute the index value instead.
2411 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2416 ex e = indexed(A, mu_co);
2417 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2418 // -> A.mu becomes A.nu
2419 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2420 // -> A.mu becomes A.0
2424 As you see, with the second method only the value of the index will get
2425 substituted. Its other properties, including its dimension, remain unchanged.
2426 If you want to change the dimension of an index you have to substitute the
2427 whole index by another one with the new dimension.
2429 Finally, substituting the base expression of an indexed object works as
2434 ex e = indexed(A, mu_co);
2435 cout << e << " becomes " << e.subs(A == A+B) << endl;
2436 // -> A.mu becomes (B+A).mu
2440 @subsection Symmetries
2441 @cindex @code{symmetry} (class)
2442 @cindex @code{sy_none()}
2443 @cindex @code{sy_symm()}
2444 @cindex @code{sy_anti()}
2445 @cindex @code{sy_cycl()}
2447 Indexed objects can have certain symmetry properties with respect to their
2448 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2449 that is constructed with the helper functions
2452 symmetry sy_none(...);
2453 symmetry sy_symm(...);
2454 symmetry sy_anti(...);
2455 symmetry sy_cycl(...);
2458 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2459 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2460 represents a cyclic symmetry. Each of these functions accepts up to four
2461 arguments which can be either symmetry objects themselves or unsigned integer
2462 numbers that represent an index position (counting from 0). A symmetry
2463 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2464 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2467 Here are some examples of symmetry definitions:
2472 e = indexed(A, i, j);
2473 e = indexed(A, sy_none(), i, j); // equivalent
2474 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2476 // Symmetric in all three indices:
2477 e = indexed(A, sy_symm(), i, j, k);
2478 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2479 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2480 // different canonical order
2482 // Symmetric in the first two indices only:
2483 e = indexed(A, sy_symm(0, 1), i, j, k);
2484 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2486 // Antisymmetric in the first and last index only (index ranges need not
2488 e = indexed(A, sy_anti(0, 2), i, j, k);
2489 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2491 // An example of a mixed symmetry: antisymmetric in the first two and
2492 // last two indices, symmetric when swapping the first and last index
2493 // pairs (like the Riemann curvature tensor):
2494 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2496 // Cyclic symmetry in all three indices:
2497 e = indexed(A, sy_cycl(), i, j, k);
2498 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2500 // The following examples are invalid constructions that will throw
2501 // an exception at run time.
2503 // An index may not appear multiple times:
2504 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2505 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2507 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2508 // same number of indices:
2509 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2511 // And of course, you cannot specify indices which are not there:
2512 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2516 If you need to specify more than four indices, you have to use the
2517 @code{.add()} method of the @code{symmetry} class. For example, to specify
2518 full symmetry in the first six indices you would write
2519 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2521 If an indexed object has a symmetry, GiNaC will automatically bring the
2522 indices into a canonical order which allows for some immediate simplifications:
2526 cout << indexed(A, sy_symm(), i, j)
2527 + indexed(A, sy_symm(), j, i) << endl;
2529 cout << indexed(B, sy_anti(), i, j)
2530 + indexed(B, sy_anti(), j, i) << endl;
2532 cout << indexed(B, sy_anti(), i, j, k)
2533 - indexed(B, sy_anti(), j, k, i) << endl;
2538 @cindex @code{get_free_indices()}
2540 @subsection Dummy indices
2542 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2543 that a summation over the index range is implied. Symbolic indices which are
2544 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2545 dummy nor free indices.
2547 To be recognized as a dummy index pair, the two indices must be of the same
2548 class and their value must be the same single symbol (an index like
2549 @samp{2*n+1} is never a dummy index). If the indices are of class
2550 @code{varidx} they must also be of opposite variance; if they are of class
2551 @code{spinidx} they must be both dotted or both undotted.
2553 The method @code{.get_free_indices()} returns a vector containing the free
2554 indices of an expression. It also checks that the free indices of the terms
2555 of a sum are consistent:
2559 symbol A("A"), B("B"), C("C");
2561 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2562 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2564 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2565 cout << exprseq(e.get_free_indices()) << endl;
2567 // 'j' and 'l' are dummy indices
2569 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2570 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2572 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2573 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2574 cout << exprseq(e.get_free_indices()) << endl;
2576 // 'nu' is a dummy index, but 'sigma' is not
2578 e = indexed(A, mu, mu);
2579 cout << exprseq(e.get_free_indices()) << endl;
2581 // 'mu' is not a dummy index because it appears twice with the same
2584 e = indexed(A, mu, nu) + 42;
2585 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2586 // this will throw an exception:
2587 // "add::get_free_indices: inconsistent indices in sum"
2591 @cindex @code{expand_dummy_sum()}
2592 A dummy index summation like
2599 can be expanded for indices with numeric
2600 dimensions (e.g. 3) into the explicit sum like
2602 $a_1b^1+a_2b^2+a_3b^3 $.
2605 a.1 b~1 + a.2 b~2 + a.3 b~3.
2607 This is performed by the function
2610 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2613 which takes an expression @code{e} and returns the expanded sum for all
2614 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2615 is set to @code{true} then all substitutions are made by @code{idx} class
2616 indices, i.e. without variance. In this case the above sum
2625 $a_1b_1+a_2b_2+a_3b_3 $.
2628 a.1 b.1 + a.2 b.2 + a.3 b.3.
2632 @cindex @code{simplify_indexed()}
2633 @subsection Simplifying indexed expressions
2635 In addition to the few automatic simplifications that GiNaC performs on
2636 indexed expressions (such as re-ordering the indices of symmetric tensors
2637 and calculating traces and convolutions of matrices and predefined tensors)
2641 ex ex::simplify_indexed();
2642 ex ex::simplify_indexed(const scalar_products & sp);
2645 that performs some more expensive operations:
2648 @item it checks the consistency of free indices in sums in the same way
2649 @code{get_free_indices()} does
2650 @item it tries to give dummy indices that appear in different terms of a sum
2651 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2652 @item it (symbolically) calculates all possible dummy index summations/contractions
2653 with the predefined tensors (this will be explained in more detail in the
2655 @item it detects contractions that vanish for symmetry reasons, for example
2656 the contraction of a symmetric and a totally antisymmetric tensor
2657 @item as a special case of dummy index summation, it can replace scalar products
2658 of two tensors with a user-defined value
2661 The last point is done with the help of the @code{scalar_products} class
2662 which is used to store scalar products with known values (this is not an
2663 arithmetic class, you just pass it to @code{simplify_indexed()}):
2667 symbol A("A"), B("B"), C("C"), i_sym("i");
2671 sp.add(A, B, 0); // A and B are orthogonal
2672 sp.add(A, C, 0); // A and C are orthogonal
2673 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2675 e = indexed(A + B, i) * indexed(A + C, i);
2677 // -> (B+A).i*(A+C).i
2679 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2685 The @code{scalar_products} object @code{sp} acts as a storage for the
2686 scalar products added to it with the @code{.add()} method. This method
2687 takes three arguments: the two expressions of which the scalar product is
2688 taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
2689 @code{simplify_indexed()} will replace all scalar products of indexed
2690 objects that have the symbols @code{A} and @code{B} as base expressions
2691 with the single value 0. The number, type and dimension of the indices
2692 don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
2694 @cindex @code{expand()}
2695 The example above also illustrates a feature of the @code{expand()} method:
2696 if passed the @code{expand_indexed} option it will distribute indices
2697 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2699 @cindex @code{tensor} (class)
2700 @subsection Predefined tensors
2702 Some frequently used special tensors such as the delta, epsilon and metric
2703 tensors are predefined in GiNaC. They have special properties when
2704 contracted with other tensor expressions and some of them have constant
2705 matrix representations (they will evaluate to a number when numeric
2706 indices are specified).
2708 @cindex @code{delta_tensor()}
2709 @subsubsection Delta tensor
2711 The delta tensor takes two indices, is symmetric and has the matrix
2712 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2713 @code{delta_tensor()}:
2717 symbol A("A"), B("B");
2719 idx i(symbol("i"), 3), j(symbol("j"), 3),
2720 k(symbol("k"), 3), l(symbol("l"), 3);
2722 ex e = indexed(A, i, j) * indexed(B, k, l)
2723 * delta_tensor(i, k) * delta_tensor(j, l);
2724 cout << e.simplify_indexed() << endl;
2727 cout << delta_tensor(i, i) << endl;
2732 @cindex @code{metric_tensor()}
2733 @subsubsection General metric tensor
2735 The function @code{metric_tensor()} creates a general symmetric metric
2736 tensor with two indices that can be used to raise/lower tensor indices. The
2737 metric tensor is denoted as @samp{g} in the output and if its indices are of
2738 mixed variance it is automatically replaced by a delta tensor:
2744 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2746 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2747 cout << e.simplify_indexed() << endl;
2750 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2751 cout << e.simplify_indexed() << endl;
2754 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2755 * metric_tensor(nu, rho);
2756 cout << e.simplify_indexed() << endl;
2759 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2760 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2761 + indexed(A, mu.toggle_variance(), rho));
2762 cout << e.simplify_indexed() << endl;
2767 @cindex @code{lorentz_g()}
2768 @subsubsection Minkowski metric tensor
2770 The Minkowski metric tensor is a special metric tensor with a constant
2771 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2772 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2773 It is created with the function @code{lorentz_g()} (although it is output as
2778 varidx mu(symbol("mu"), 4);
2780 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2781 * lorentz_g(mu, varidx(0, 4)); // negative signature
2782 cout << e.simplify_indexed() << endl;
2785 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2786 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2787 cout << e.simplify_indexed() << endl;
2792 @cindex @code{spinor_metric()}
2793 @subsubsection Spinor metric tensor
2795 The function @code{spinor_metric()} creates an antisymmetric tensor with
2796 two indices that is used to raise/lower indices of 2-component spinors.
2797 It is output as @samp{eps}:
2803 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2804 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2806 e = spinor_metric(A, B) * indexed(psi, B_co);
2807 cout << e.simplify_indexed() << endl;
2810 e = spinor_metric(A, B) * indexed(psi, A_co);
2811 cout << e.simplify_indexed() << endl;
2814 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2815 cout << e.simplify_indexed() << endl;
2818 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2819 cout << e.simplify_indexed() << endl;
2822 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2823 cout << e.simplify_indexed() << endl;
2826 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2827 cout << e.simplify_indexed() << endl;
2832 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2834 @cindex @code{epsilon_tensor()}
2835 @cindex @code{lorentz_eps()}
2836 @subsubsection Epsilon tensor
2838 The epsilon tensor is totally antisymmetric, its number of indices is equal
2839 to the dimension of the index space (the indices must all be of the same
2840 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2841 defined to be 1. Its behavior with indices that have a variance also
2842 depends on the signature of the metric. Epsilon tensors are output as
2845 There are three functions defined to create epsilon tensors in 2, 3 and 4
2849 ex epsilon_tensor(const ex & i1, const ex & i2);
2850 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2851 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2852 bool pos_sig = false);
2855 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2856 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2857 Minkowski space (the last @code{bool} argument specifies whether the metric
2858 has negative or positive signature, as in the case of the Minkowski metric
2863 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2864 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2865 e = lorentz_eps(mu, nu, rho, sig) *
2866 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2867 cout << simplify_indexed(e) << endl;
2868 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2870 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2871 symbol A("A"), B("B");
2872 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2873 cout << simplify_indexed(e) << endl;
2874 // -> -B.k*A.j*eps.i.k.j
2875 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2876 cout << simplify_indexed(e) << endl;
2881 @subsection Linear algebra
2883 The @code{matrix} class can be used with indices to do some simple linear
2884 algebra (linear combinations and products of vectors and matrices, traces
2885 and scalar products):
2889 idx i(symbol("i"), 2), j(symbol("j"), 2);
2890 symbol x("x"), y("y");
2892 // A is a 2x2 matrix, X is a 2x1 vector
2893 matrix A(2, 2), X(2, 1);
2898 cout << indexed(A, i, i) << endl;
2901 ex e = indexed(A, i, j) * indexed(X, j);
2902 cout << e.simplify_indexed() << endl;
2903 // -> [[2*y+x],[4*y+3*x]].i
2905 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2906 cout << e.simplify_indexed() << endl;
2907 // -> [[3*y+3*x,6*y+2*x]].j
2911 You can of course obtain the same results with the @code{matrix::add()},
2912 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2913 but with indices you don't have to worry about transposing matrices.
2915 Matrix indices always start at 0 and their dimension must match the number
2916 of rows/columns of the matrix. Matrices with one row or one column are
2917 vectors and can have one or two indices (it doesn't matter whether it's a
2918 row or a column vector). Other matrices must have two indices.
2920 You should be careful when using indices with variance on matrices. GiNaC
2921 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2922 @samp{F.mu.nu} are different matrices. In this case you should use only
2923 one form for @samp{F} and explicitly multiply it with a matrix representation
2924 of the metric tensor.
2927 @node Non-commutative objects, Hash Maps, Indexed objects, Basic Concepts
2928 @c node-name, next, previous, up
2929 @section Non-commutative objects
2931 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2932 non-commutative objects are built-in which are mostly of use in high energy
2936 @item Clifford (Dirac) algebra (class @code{clifford})
2937 @item su(3) Lie algebra (class @code{color})
2938 @item Matrices (unindexed) (class @code{matrix})
2941 The @code{clifford} and @code{color} classes are subclasses of
2942 @code{indexed} because the elements of these algebras usually carry
2943 indices. The @code{matrix} class is described in more detail in
2946 Unlike most computer algebra systems, GiNaC does not primarily provide an
2947 operator (often denoted @samp{&*}) for representing inert products of
2948 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2949 classes of objects involved, and non-commutative products are formed with
2950 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2951 figuring out by itself which objects commutate and will group the factors
2952 by their class. Consider this example:
2956 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2957 idx a(symbol("a"), 8), b(symbol("b"), 8);
2958 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2960 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
2964 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
2965 groups the non-commutative factors (the gammas and the su(3) generators)
2966 together while preserving the order of factors within each class (because
2967 Clifford objects commutate with color objects). The resulting expression is a
2968 @emph{commutative} product with two factors that are themselves non-commutative
2969 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
2970 parentheses are placed around the non-commutative products in the output.
2972 @cindex @code{ncmul} (class)
2973 Non-commutative products are internally represented by objects of the class
2974 @code{ncmul}, as opposed to commutative products which are handled by the
2975 @code{mul} class. You will normally not have to worry about this distinction,
2978 The advantage of this approach is that you never have to worry about using
2979 (or forgetting to use) a special operator when constructing non-commutative
2980 expressions. Also, non-commutative products in GiNaC are more intelligent
2981 than in other computer algebra systems; they can, for example, automatically
2982 canonicalize themselves according to rules specified in the implementation
2983 of the non-commutative classes. The drawback is that to work with other than
2984 the built-in algebras you have to implement new classes yourself. Symbols
2985 always commutate and it's not possible to construct non-commutative products
2986 using symbols to represent the algebra elements or generators. User-defined
2987 functions can, however, be specified as being non-commutative.
2989 @cindex @code{return_type()}
2990 @cindex @code{return_type_tinfo()}
2991 Information about the commutativity of an object or expression can be
2992 obtained with the two member functions
2995 unsigned ex::return_type() const;
2996 unsigned ex::return_type_tinfo() const;
2999 The @code{return_type()} function returns one of three values (defined in
3000 the header file @file{flags.h}), corresponding to three categories of
3001 expressions in GiNaC:
3004 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3005 classes are of this kind.
3006 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3007 certain class of non-commutative objects which can be determined with the
3008 @code{return_type_tinfo()} method. Expressions of this category commutate
3009 with everything except @code{noncommutative} expressions of the same
3011 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3012 of non-commutative objects of different classes. Expressions of this
3013 category don't commutate with any other @code{noncommutative} or
3014 @code{noncommutative_composite} expressions.
3017 The value returned by the @code{return_type_tinfo()} method is valid only
3018 when the return type of the expression is @code{noncommutative}. It is a
3019 value that is unique to the class of the object and usually one of the
3020 constants in @file{tinfos.h}, or derived therefrom.
3022 Here are a couple of examples:
3025 @multitable @columnfractions 0.33 0.33 0.34
3026 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3027 @item @code{42} @tab @code{commutative} @tab -
3028 @item @code{2*x-y} @tab @code{commutative} @tab -
3029 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3030 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3031 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3032 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3036 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3037 @code{TINFO_clifford} for objects with a representation label of zero.
3038 Other representation labels yield a different @code{return_type_tinfo()},
3039 but it's the same for any two objects with the same label. This is also true
3042 A last note: With the exception of matrices, positive integer powers of
3043 non-commutative objects are automatically expanded in GiNaC. For example,
3044 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3045 non-commutative expressions).
3048 @cindex @code{clifford} (class)
3049 @subsection Clifford algebra
3052 Clifford algebras are supported in two flavours: Dirac gamma
3053 matrices (more physical) and generic Clifford algebras (more
3056 @cindex @code{dirac_gamma()}
3057 @subsubsection Dirac gamma matrices
3058 Dirac gamma matrices (note that GiNaC doesn't treat them
3059 as matrices) are designated as @samp{gamma~mu} and satisfy
3060 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3061 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3062 constructed by the function
3065 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3068 which takes two arguments: the index and a @dfn{representation label} in the
3069 range 0 to 255 which is used to distinguish elements of different Clifford
3070 algebras (this is also called a @dfn{spin line index}). Gammas with different
3071 labels commutate with each other. The dimension of the index can be 4 or (in
3072 the framework of dimensional regularization) any symbolic value. Spinor
3073 indices on Dirac gammas are not supported in GiNaC.
3075 @cindex @code{dirac_ONE()}
3076 The unity element of a Clifford algebra is constructed by
3079 ex dirac_ONE(unsigned char rl = 0);
3082 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3083 multiples of the unity element, even though it's customary to omit it.
3084 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3085 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3086 GiNaC will complain and/or produce incorrect results.
3088 @cindex @code{dirac_gamma5()}
3089 There is a special element @samp{gamma5} that commutates with all other
3090 gammas, has a unit square, and in 4 dimensions equals
3091 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3094 ex dirac_gamma5(unsigned char rl = 0);
3097 @cindex @code{dirac_gammaL()}
3098 @cindex @code{dirac_gammaR()}
3099 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3100 objects, constructed by
3103 ex dirac_gammaL(unsigned char rl = 0);
3104 ex dirac_gammaR(unsigned char rl = 0);
3107 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3108 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3110 @cindex @code{dirac_slash()}
3111 Finally, the function
3114 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3117 creates a term that represents a contraction of @samp{e} with the Dirac
3118 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3119 with a unique index whose dimension is given by the @code{dim} argument).
3120 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3122 In products of dirac gammas, superfluous unity elements are automatically
3123 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3124 and @samp{gammaR} are moved to the front.
3126 The @code{simplify_indexed()} function performs contractions in gamma strings,
3132 symbol a("a"), b("b"), D("D");
3133 varidx mu(symbol("mu"), D);
3134 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3135 * dirac_gamma(mu.toggle_variance());
3137 // -> gamma~mu*a\*gamma.mu
3138 e = e.simplify_indexed();
3141 cout << e.subs(D == 4) << endl;
3147 @cindex @code{dirac_trace()}
3148 To calculate the trace of an expression containing strings of Dirac gammas
3149 you use one of the functions
3152 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3153 const ex & trONE = 4);
3154 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3155 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3158 These functions take the trace over all gammas in the specified set @code{rls}
3159 or list @code{rll} of representation labels, or the single label @code{rl};
3160 gammas with other labels are left standing. The last argument to
3161 @code{dirac_trace()} is the value to be returned for the trace of the unity
3162 element, which defaults to 4.
3164 The @code{dirac_trace()} function is a linear functional that is equal to the
3165 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3166 functional is not cyclic in
3169 dimensions when acting on
3170 expressions containing @samp{gamma5}, so it's not a proper trace. This
3171 @samp{gamma5} scheme is described in greater detail in
3172 @cite{The Role of gamma5 in Dimensional Regularization}.
3174 The value of the trace itself is also usually different in 4 and in
3182 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3183 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3184 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3185 cout << dirac_trace(e).simplify_indexed() << endl;
3192 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3193 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3194 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3195 cout << dirac_trace(e).simplify_indexed() << endl;
3196 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3200 Here is an example for using @code{dirac_trace()} to compute a value that
3201 appears in the calculation of the one-loop vacuum polarization amplitude in
3206 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3207 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3210 sp.add(l, l, pow(l, 2));
3211 sp.add(l, q, ldotq);
3213 ex e = dirac_gamma(mu) *
3214 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3215 dirac_gamma(mu.toggle_variance()) *
3216 (dirac_slash(l, D) + m * dirac_ONE());
3217 e = dirac_trace(e).simplify_indexed(sp);
3218 e = e.collect(lst(l, ldotq, m));
3220 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3224 The @code{canonicalize_clifford()} function reorders all gamma products that
3225 appear in an expression to a canonical (but not necessarily simple) form.
3226 You can use this to compare two expressions or for further simplifications:
3230 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3231 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3233 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3235 e = canonicalize_clifford(e);
3237 // -> 2*ONE*eta~mu~nu
3241 @cindex @code{clifford_unit()}
3242 @subsubsection A generic Clifford algebra
3244 A generic Clifford algebra, i.e. a
3248 dimensional algebra with
3252 satisfying the identities
3254 $e_i e_j + e_j e_i = M(i, j) + M(j, i) $
3257 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3259 for some bilinear form (@code{metric})
3260 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3261 and contain symbolic entries. Such generators are created by the
3265 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0,
3266 bool anticommuting = false);
3269 where @code{mu} should be a @code{varidx} class object indexing the
3270 generators, an index @code{mu} with a numeric value may be of type
3272 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3273 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3274 object. Optional parameter @code{rl} allows to distinguish different
3275 Clifford algebras, which will commute with each other. The last
3276 optional parameter @code{anticommuting} defines if the anticommuting
3279 $e_i e_j + e_j e_i = 0$)
3282 e~i e~j + e~j e~i = 0)
3284 will be used for contraction of Clifford units. If the @code{metric} is
3285 supplied by a @code{matrix} object, then the value of
3286 @code{anticommuting} is calculated automatically and the supplied one
3287 will be ignored. One can overcome this by giving @code{metric} through
3288 matrix wrapped into an @code{indexed} object.
3290 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3291 something very close to @code{dirac_gamma(mu)}, although
3292 @code{dirac_gamma} have more efficient simplification mechanism.
3293 @cindex @code{clifford::get_metric()}
3294 The method @code{clifford::get_metric()} returns a metric defining this
3296 @cindex @code{clifford::is_anticommuting()}
3297 The method @code{clifford::is_anticommuting()} returns the
3298 @code{anticommuting} property of a unit.
3300 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3301 the Clifford algebra units with a call like that
3304 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3307 since this may yield some further automatic simplifications. Again, for a
3308 metric defined through a @code{matrix} such a symmetry is detected
3311 Individual generators of a Clifford algebra can be accessed in several
3317 varidx nu(symbol("nu"), 4);
3319 ex M = diag_matrix(lst(1, -1, 0, s));
3320 ex e = clifford_unit(nu, M);
3321 ex e0 = e.subs(nu == 0);
3322 ex e1 = e.subs(nu == 1);
3323 ex e2 = e.subs(nu == 2);
3324 ex e3 = e.subs(nu == 3);
3329 will produce four anti-commuting generators of a Clifford algebra with properties
3331 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3334 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3335 @code{pow(e3, 2) = s}.
3338 @cindex @code{lst_to_clifford()}
3339 A similar effect can be achieved from the function
3342 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3343 unsigned char rl = 0, bool anticommuting = false);
3344 ex lst_to_clifford(const ex & v, const ex & e);
3347 which converts a list or vector
3349 $v = (v^0, v^1, ..., v^n)$
3352 @samp{v = (v~0, v~1, ..., v~n)}
3357 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3360 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3363 directly supplied in the second form of the procedure. In the first form
3364 the Clifford unit @samp{e.k} is generated by the call of
3365 @code{clifford_unit(mu, metr, rl, anticommuting)}. The previous code may be rewritten
3366 with the help of @code{lst_to_clifford()} as follows
3371 varidx nu(symbol("nu"), 4);
3373 ex M = diag_matrix(lst(1, -1, 0, s));
3374 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3375 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3376 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3377 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3382 @cindex @code{clifford_to_lst()}
3383 There is the inverse function
3386 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3389 which takes an expression @code{e} and tries to find a list
3391 $v = (v^0, v^1, ..., v^n)$
3394 @samp{v = (v~0, v~1, ..., v~n)}
3398 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3401 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3403 with respect to the given Clifford units @code{c} and with none of the
3404 @samp{v~k} containing Clifford units @code{c} (of course, this
3405 may be impossible). This function can use an @code{algebraic} method
3406 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3408 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3411 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3413 is zero or is not @code{numeric} for some @samp{k}
3414 then the method will be automatically changed to symbolic. The same effect
3415 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3417 @cindex @code{clifford_prime()}
3418 @cindex @code{clifford_star()}
3419 @cindex @code{clifford_bar()}
3420 There are several functions for (anti-)automorphisms of Clifford algebras:
3423 ex clifford_prime(const ex & e)
3424 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3425 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3428 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3429 changes signs of all Clifford units in the expression. The reversion
3430 of a Clifford algebra @code{clifford_star()} coincides with the
3431 @code{conjugate()} method and effectively reverses the order of Clifford
3432 units in any product. Finally the main anti-automorphism
3433 of a Clifford algebra @code{clifford_bar()} is the composition of the
3434 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3435 in a product. These functions correspond to the notations
3450 used in Clifford algebra textbooks.
3452 @cindex @code{clifford_norm()}
3456 ex clifford_norm(const ex & e);
3459 @cindex @code{clifford_inverse()}
3460 calculates the norm of a Clifford number from the expression
3462 $||e||^2 = e\overline{e}$.
3465 @code{||e||^2 = e \bar@{e@}}
3467 The inverse of a Clifford expression is returned by the function
3470 ex clifford_inverse(const ex & e);
3473 which calculates it as
3475 $e^{-1} = \overline{e}/||e||^2$.
3478 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3487 then an exception is raised.
3489 @cindex @code{remove_dirac_ONE()}
3490 If a Clifford number happens to be a factor of
3491 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3492 expression by the function
3495 ex remove_dirac_ONE(const ex & e);
3498 @cindex @code{canonicalize_clifford()}
3499 The function @code{canonicalize_clifford()} works for a
3500 generic Clifford algebra in a similar way as for Dirac gammas.
3502 The next provided function is
3504 @cindex @code{clifford_moebius_map()}
3506 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3507 const ex & d, const ex & v, const ex & G,
3508 unsigned char rl = 0, bool anticommuting = false);
3509 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3510 unsigned char rl = 0, bool anticommuting = false);
3513 It takes a list or vector @code{v} and makes the Moebius (conformal or
3514 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3515 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3516 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3517 indexed object, tensormetric, matrix or a Clifford unit, in the later
3518 case the optional parameters @code{rl} and @code{anticommuting} are ignored
3519 even if supplied. The returned value of this function is a list of
3520 components of the resulting vector.
3522 @cindex @code{clifford_max_label()}
3523 Finally the function
3526 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3529 can detect a presence of Clifford objects in the expression @code{e}: if
3530 such objects are found it returns the maximal
3531 @code{representation_label} of them, otherwise @code{-1}. The optional
3532 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3533 be ignored during the search.
3535 LaTeX output for Clifford units looks like
3536 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3537 @code{representation_label} and @code{\nu} is the index of the
3538 corresponding unit. This provides a flexible typesetting with a suitable
3539 defintion of the @code{\clifford} command. For example, the definition
3541 \newcommand@{\clifford@}[1][]@{@}
3543 typesets all Clifford units identically, while the alternative definition
3545 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3547 prints units with @code{representation_label=0} as
3554 with @code{representation_label=1} as
3561 and with @code{representation_label=2} as
3569 @cindex @code{color} (class)
3570 @subsection Color algebra
3572 @cindex @code{color_T()}
3573 For computations in quantum chromodynamics, GiNaC implements the base elements
3574 and structure constants of the su(3) Lie algebra (color algebra). The base
3575 elements @math{T_a} are constructed by the function
3578 ex color_T(const ex & a, unsigned char rl = 0);
3581 which takes two arguments: the index and a @dfn{representation label} in the
3582 range 0 to 255 which is used to distinguish elements of different color
3583 algebras. Objects with different labels commutate with each other. The
3584 dimension of the index must be exactly 8 and it should be of class @code{idx},
3587 @cindex @code{color_ONE()}
3588 The unity element of a color algebra is constructed by
3591 ex color_ONE(unsigned char rl = 0);
3594 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3595 multiples of the unity element, even though it's customary to omit it.
3596 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3597 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3598 GiNaC may produce incorrect results.
3600 @cindex @code{color_d()}
3601 @cindex @code{color_f()}
3605 ex color_d(const ex & a, const ex & b, const ex & c);
3606 ex color_f(const ex & a, const ex & b, const ex & c);
3609 create the symmetric and antisymmetric structure constants @math{d_abc} and
3610 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3611 and @math{[T_a, T_b] = i f_abc T_c}.
3613 These functions evaluate to their numerical values,
3614 if you supply numeric indices to them. The index values should be in
3615 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3616 goes along better with the notations used in physical literature.
3618 @cindex @code{color_h()}
3619 There's an additional function
3622 ex color_h(const ex & a, const ex & b, const ex & c);
3625 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3627 The function @code{simplify_indexed()} performs some simplifications on
3628 expressions containing color objects:
3633 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3634 k(symbol("k"), 8), l(symbol("l"), 8);
3636 e = color_d(a, b, l) * color_f(a, b, k);
3637 cout << e.simplify_indexed() << endl;
3640 e = color_d(a, b, l) * color_d(a, b, k);
3641 cout << e.simplify_indexed() << endl;
3644 e = color_f(l, a, b) * color_f(a, b, k);
3645 cout << e.simplify_indexed() << endl;
3648 e = color_h(a, b, c) * color_h(a, b, c);
3649 cout << e.simplify_indexed() << endl;
3652 e = color_h(a, b, c) * color_T(b) * color_T(c);
3653 cout << e.simplify_indexed() << endl;
3656 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3657 cout << e.simplify_indexed() << endl;
3660 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3661 cout << e.simplify_indexed() << endl;
3662 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3666 @cindex @code{color_trace()}
3667 To calculate the trace of an expression containing color objects you use one
3671 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3672 ex color_trace(const ex & e, const lst & rll);
3673 ex color_trace(const ex & e, unsigned char rl = 0);
3676 These functions take the trace over all color @samp{T} objects in the
3677 specified set @code{rls} or list @code{rll} of representation labels, or the
3678 single label @code{rl}; @samp{T}s with other labels are left standing. For
3683 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3685 // -> -I*f.a.c.b+d.a.c.b
3690 @node Hash Maps, Methods and Functions, Non-commutative objects, Basic Concepts
3691 @c node-name, next, previous, up
3694 @cindex @code{exhashmap} (class)
3696 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3697 that can be used as a drop-in replacement for the STL
3698 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3699 typically constant-time, element look-up than @code{map<>}.
3701 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3702 following differences:
3706 no @code{lower_bound()} and @code{upper_bound()} methods
3708 no reverse iterators, no @code{rbegin()}/@code{rend()}
3710 no @code{operator<(exhashmap, exhashmap)}
3712 the comparison function object @code{key_compare} is hardcoded to
3715 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3716 initial hash table size (the actual table size after construction may be
3717 larger than the specified value)
3719 the method @code{size_t bucket_count()} returns the current size of the hash
3722 @code{insert()} and @code{erase()} operations invalidate all iterators
3726 @node Methods and Functions, Information About Expressions, Hash Maps, Top
3727 @c node-name, next, previous, up
3728 @chapter Methods and Functions
3731 In this chapter the most important algorithms provided by GiNaC will be
3732 described. Some of them are implemented as functions on expressions,
3733 others are implemented as methods provided by expression objects. If
3734 they are methods, there exists a wrapper function around it, so you can
3735 alternatively call it in a functional way as shown in the simple
3740 cout << "As method: " << sin(1).evalf() << endl;
3741 cout << "As function: " << evalf(sin(1)) << endl;
3745 @cindex @code{subs()}
3746 The general rule is that wherever methods accept one or more parameters
3747 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3748 wrapper accepts is the same but preceded by the object to act on
3749 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3750 most natural one in an OO model but it may lead to confusion for MapleV
3751 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3752 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3753 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3754 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3755 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3756 here. Also, users of MuPAD will in most cases feel more comfortable
3757 with GiNaC's convention. All function wrappers are implemented
3758 as simple inline functions which just call the corresponding method and
3759 are only provided for users uncomfortable with OO who are dead set to
3760 avoid method invocations. Generally, nested function wrappers are much
3761 harder to read than a sequence of methods and should therefore be
3762 avoided if possible. On the other hand, not everything in GiNaC is a
3763 method on class @code{ex} and sometimes calling a function cannot be
3767 * Information About Expressions::
3768 * Numerical Evaluation::
3769 * Substituting Expressions::
3770 * Pattern Matching and Advanced Substitutions::
3771 * Applying a Function on Subexpressions::
3772 * Visitors and Tree Traversal::
3773 * Polynomial Arithmetic:: Working with polynomials.
3774 * Rational Expressions:: Working with rational functions.
3775 * Symbolic Differentiation::
3776 * Series Expansion:: Taylor and Laurent expansion.
3778 * Built-in Functions:: List of predefined mathematical functions.
3779 * Multiple polylogarithms::
3780 * Complex Conjugation::
3781 * Built-in Functions:: List of predefined mathematical functions.
3782 * Solving Linear Systems of Equations::
3783 * Input/Output:: Input and output of expressions.
3787 @node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
3788 @c node-name, next, previous, up
3789 @section Getting information about expressions
3791 @subsection Checking expression types
3792 @cindex @code{is_a<@dots{}>()}
3793 @cindex @code{is_exactly_a<@dots{}>()}
3794 @cindex @code{ex_to<@dots{}>()}
3795 @cindex Converting @code{ex} to other classes
3796 @cindex @code{info()}
3797 @cindex @code{return_type()}
3798 @cindex @code{return_type_tinfo()}
3800 Sometimes it's useful to check whether a given expression is a plain number,
3801 a sum, a polynomial with integer coefficients, or of some other specific type.
3802 GiNaC provides a couple of functions for this:
3805 bool is_a<T>(const ex & e);
3806 bool is_exactly_a<T>(const ex & e);
3807 bool ex::info(unsigned flag);
3808 unsigned ex::return_type() const;
3809 unsigned ex::return_type_tinfo() const;
3812 When the test made by @code{is_a<T>()} returns true, it is safe to call
3813 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3814 class names (@xref{The Class Hierarchy}, for a list of all classes). For
3815 example, assuming @code{e} is an @code{ex}:
3820 if (is_a<numeric>(e))
3821 numeric n = ex_to<numeric>(e);
3826 @code{is_a<T>(e)} allows you to check whether the top-level object of
3827 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3828 (@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
3829 e.g., for checking whether an expression is a number, a sum, or a product:
3836 is_a<numeric>(e1); // true
3837 is_a<numeric>(e2); // false
3838 is_a<add>(e1); // false
3839 is_a<add>(e2); // true
3840 is_a<mul>(e1); // false
3841 is_a<mul>(e2); // false
3845 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3846 top-level object of an expression @samp{e} is an instance of the GiNaC
3847 class @samp{T}, not including parent classes.
3849 The @code{info()} method is used for checking certain attributes of
3850 expressions. The possible values for the @code{flag} argument are defined
3851 in @file{ginac/flags.h}, the most important being explained in the following
3855 @multitable @columnfractions .30 .70
3856 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3857 @item @code{numeric}
3858 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3860 @tab @dots{}a real integer, rational or float (i.e. is not complex)
3861 @item @code{rational}
3862 @tab @dots{}an exact rational number (integers are rational, too)
3863 @item @code{integer}
3864 @tab @dots{}a (non-complex) integer
3865 @item @code{crational}
3866 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3867 @item @code{cinteger}
3868 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3869 @item @code{positive}
3870 @tab @dots{}not complex and greater than 0
3871 @item @code{negative}
3872 @tab @dots{}not complex and less than 0
3873 @item @code{nonnegative}
3874 @tab @dots{}not complex and greater than or equal to 0
3876 @tab @dots{}an integer greater than 0
3878 @tab @dots{}an integer less than 0
3879 @item @code{nonnegint}
3880 @tab @dots{}an integer greater than or equal to 0
3882 @tab @dots{}an even integer
3884 @tab @dots{}an odd integer
3886 @tab @dots{}a prime integer (probabilistic primality test)
3887 @item @code{relation}
3888 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3889 @item @code{relation_equal}
3890 @tab @dots{}a @code{==} relation
3891 @item @code{relation_not_equal}
3892 @tab @dots{}a @code{!=} relation
3893 @item @code{relation_less}
3894 @tab @dots{}a @code{<} relation
3895 @item @code{relation_less_or_equal}
3896 @tab @dots{}a @code{<=} relation
3897 @item @code{relation_greater}
3898 @tab @dots{}a @code{>} relation
3899 @item @code{relation_greater_or_equal}
3900 @tab @dots{}a @code{>=} relation
3902 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3904 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3905 @item @code{polynomial}
3906 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3907 @item @code{integer_polynomial}
3908 @tab @dots{}a polynomial with (non-complex) integer coefficients
3909 @item @code{cinteger_polynomial}
3910 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3911 @item @code{rational_polynomial}
3912 @tab @dots{}a polynomial with (non-complex) rational coefficients
3913 @item @code{crational_polynomial}
3914 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3915 @item @code{rational_function}
3916 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3917 @item @code{algebraic}
3918 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3922 To determine whether an expression is commutative or non-commutative and if
3923 so, with which other expressions it would commutate, you use the methods
3924 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3925 for an explanation of these.
3928 @subsection Accessing subexpressions
3931 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3932 @code{function}, act as containers for subexpressions. For example, the
3933 subexpressions of a sum (an @code{add} object) are the individual terms,
3934 and the subexpressions of a @code{function} are the function's arguments.
3936 @cindex @code{nops()}
3938 GiNaC provides several ways of accessing subexpressions. The first way is to
3943 ex ex::op(size_t i);
3946 @code{nops()} determines the number of subexpressions (operands) contained
3947 in the expression, while @code{op(i)} returns the @code{i}-th
3948 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3949 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3950 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3951 @math{i>0} are the indices.
3954 @cindex @code{const_iterator}
3955 The second way to access subexpressions is via the STL-style random-access
3956 iterator class @code{const_iterator} and the methods
3959 const_iterator ex::begin();
3960 const_iterator ex::end();
3963 @code{begin()} returns an iterator referring to the first subexpression;
3964 @code{end()} returns an iterator which is one-past the last subexpression.
3965 If the expression has no subexpressions, then @code{begin() == end()}. These
3966 iterators can also be used in conjunction with non-modifying STL algorithms.
3968 Here is an example that (non-recursively) prints the subexpressions of a
3969 given expression in three different ways:
3976 for (size_t i = 0; i != e.nops(); ++i)
3977 cout << e.op(i) << endl;
3980 for (const_iterator i = e.begin(); i != e.end(); ++i)
3983 // with iterators and STL copy()
3984 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
3988 @cindex @code{const_preorder_iterator}
3989 @cindex @code{const_postorder_iterator}
3990 @code{op()}/@code{nops()} and @code{const_iterator} only access an
3991 expression's immediate children. GiNaC provides two additional iterator
3992 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
3993 that iterate over all objects in an expression tree, in preorder or postorder,
3994 respectively. They are STL-style forward iterators, and are created with the
3998 const_preorder_iterator ex::preorder_begin();
3999 const_preorder_iterator ex::preorder_end();
4000 const_postorder_iterator ex::postorder_begin();
4001 const_postorder_iterator ex::postorder_end();
4004 The following example illustrates the differences between
4005 @code{const_iterator}, @code{const_preorder_iterator}, and
4006 @code{const_postorder_iterator}:
4010 symbol A("A"), B("B"), C("C");
4011 ex e = lst(lst(A, B), C);
4013 std::copy(e.begin(), e.end(),
4014 std::ostream_iterator<ex>(cout, "\n"));
4018 std::copy(e.preorder_begin(), e.preorder_end(),
4019 std::ostream_iterator<ex>(cout, "\n"));
4026 std::copy(e.postorder_begin(), e.postorder_end(),
4027 std::ostream_iterator<ex>(cout, "\n"));
4036 @cindex @code{relational} (class)
4037 Finally, the left-hand side and right-hand side expressions of objects of
4038 class @code{relational} (and only of these) can also be accessed with the
4047 @subsection Comparing expressions
4048 @cindex @code{is_equal()}
4049 @cindex @code{is_zero()}
4051 Expressions can be compared with the usual C++ relational operators like
4052 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4053 the result is usually not determinable and the result will be @code{false},
4054 except in the case of the @code{!=} operator. You should also be aware that
4055 GiNaC will only do the most trivial test for equality (subtracting both
4056 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4059 Actually, if you construct an expression like @code{a == b}, this will be
4060 represented by an object of the @code{relational} class (@pxref{Relations})
4061 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4063 There are also two methods
4066 bool ex::is_equal(const ex & other);
4070 for checking whether one expression is equal to another, or equal to zero,
4074 @subsection Ordering expressions
4075 @cindex @code{ex_is_less} (class)
4076 @cindex @code{ex_is_equal} (class)
4077 @cindex @code{compare()}
4079 Sometimes it is necessary to establish a mathematically well-defined ordering
4080 on a set of arbitrary expressions, for example to use expressions as keys
4081 in a @code{std::map<>} container, or to bring a vector of expressions into
4082 a canonical order (which is done internally by GiNaC for sums and products).
4084 The operators @code{<}, @code{>} etc. described in the last section cannot
4085 be used for this, as they don't implement an ordering relation in the
4086 mathematical sense. In particular, they are not guaranteed to be
4087 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4088 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4091 By default, STL classes and algorithms use the @code{<} and @code{==}
4092 operators to compare objects, which are unsuitable for expressions, but GiNaC
4093 provides two functors that can be supplied as proper binary comparison
4094 predicates to the STL:
4097 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4099 bool operator()(const ex &lh, const ex &rh) const;
4102 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4104 bool operator()(const ex &lh, const ex &rh) const;
4108 For example, to define a @code{map} that maps expressions to strings you
4112 std::map<ex, std::string, ex_is_less> myMap;
4115 Omitting the @code{ex_is_less} template parameter will introduce spurious
4116 bugs because the map operates improperly.
4118 Other examples for the use of the functors:
4126 std::sort(v.begin(), v.end(), ex_is_less());
4128 // count the number of expressions equal to '1'
4129 unsigned num_ones = std::count_if(v.begin(), v.end(),
4130 std::bind2nd(ex_is_equal(), 1));
4133 The implementation of @code{ex_is_less} uses the member function
4136 int ex::compare(const ex & other) const;
4139 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4140 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4144 @node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
4145 @c node-name, next, previous, up
4146 @section Numerical Evaluation
4147 @cindex @code{evalf()}
4149 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4150 To evaluate them using floating-point arithmetic you need to call
4153 ex ex::evalf(int level = 0) const;
4156 @cindex @code{Digits}
4157 The accuracy of the evaluation is controlled by the global object @code{Digits}
4158 which can be assigned an integer value. The default value of @code{Digits}
4159 is 17. @xref{Numbers}, for more information and examples.
4161 To evaluate an expression to a @code{double} floating-point number you can
4162 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4166 // Approximate sin(x/Pi)
4168 ex e = series(sin(x/Pi), x == 0, 6);
4170 // Evaluate numerically at x=0.1
4171 ex f = evalf(e.subs(x == 0.1));
4173 // ex_to<numeric> is an unsafe cast, so check the type first
4174 if (is_a<numeric>(f)) @{
4175 double d = ex_to<numeric>(f).to_double();
4184 @node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
4185 @c node-name, next, previous, up
4186 @section Substituting expressions
4187 @cindex @code{subs()}
4189 Algebraic objects inside expressions can be replaced with arbitrary
4190 expressions via the @code{.subs()} method:
4193 ex ex::subs(const ex & e, unsigned options = 0);
4194 ex ex::subs(const exmap & m, unsigned options = 0);
4195 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4198 In the first form, @code{subs()} accepts a relational of the form
4199 @samp{object == expression} or a @code{lst} of such relationals:
4203 symbol x("x"), y("y");
4205 ex e1 = 2*x^2-4*x+3;
4206 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4210 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4215 If you specify multiple substitutions, they are performed in parallel, so e.g.
4216 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4218 The second form of @code{subs()} takes an @code{exmap} object which is a
4219 pair associative container that maps expressions to expressions (currently
4220 implemented as a @code{std::map}). This is the most efficient one of the
4221 three @code{subs()} forms and should be used when the number of objects to
4222 be substituted is large or unknown.
4224 Using this form, the second example from above would look like this:
4228 symbol x("x"), y("y");
4234 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4238 The third form of @code{subs()} takes two lists, one for the objects to be
4239 replaced and one for the expressions to be substituted (both lists must
4240 contain the same number of elements). Using this form, you would write
4244 symbol x("x"), y("y");
4247 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4251 The optional last argument to @code{subs()} is a combination of
4252 @code{subs_options} flags. There are two options available:
4253 @code{subs_options::no_pattern} disables pattern matching, which makes
4254 large @code{subs()} operations significantly faster if you are not using
4255 patterns. The second option, @code{subs_options::algebraic} enables
4256 algebraic substitutions in products and powers.
4257 @ref{Pattern Matching and Advanced Substitutions}, for more information
4258 about patterns and algebraic substitutions.
4260 @code{subs()} performs syntactic substitution of any complete algebraic
4261 object; it does not try to match sub-expressions as is demonstrated by the
4266 symbol x("x"), y("y"), z("z");
4268 ex e1 = pow(x+y, 2);
4269 cout << e1.subs(x+y == 4) << endl;
4272 ex e2 = sin(x)*sin(y)*cos(x);
4273 cout << e2.subs(sin(x) == cos(x)) << endl;
4274 // -> cos(x)^2*sin(y)
4277 cout << e3.subs(x+y == 4) << endl;
4279 // (and not 4+z as one might expect)
4283 A more powerful form of substitution using wildcards is described in the
4287 @node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
4288 @c node-name, next, previous, up
4289 @section Pattern matching and advanced substitutions
4290 @cindex @code{wildcard} (class)
4291 @cindex Pattern matching
4293 GiNaC allows the use of patterns for checking whether an expression is of a
4294 certain form or contains subexpressions of a certain form, and for
4295 substituting expressions in a more general way.
4297 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4298 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4299 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4300 an unsigned integer number to allow having multiple different wildcards in a
4301 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4302 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4306 ex wild(unsigned label = 0);
4309 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4312 Some examples for patterns:
4314 @multitable @columnfractions .5 .5
4315 @item @strong{Constructed as} @tab @strong{Output as}
4316 @item @code{wild()} @tab @samp{$0}
4317 @item @code{pow(x,wild())} @tab @samp{x^$0}
4318 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4319 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4325 @item Wildcards behave like symbols and are subject to the same algebraic
4326 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4327 @item As shown in the last example, to use wildcards for indices you have to
4328 use them as the value of an @code{idx} object. This is because indices must
4329 always be of class @code{idx} (or a subclass).
4330 @item Wildcards only represent expressions or subexpressions. It is not
4331 possible to use them as placeholders for other properties like index
4332 dimension or variance, representation labels, symmetry of indexed objects
4334 @item Because wildcards are commutative, it is not possible to use wildcards
4335 as part of noncommutative products.
4336 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4337 are also valid patterns.
4340 @subsection Matching expressions
4341 @cindex @code{match()}
4342 The most basic application of patterns is to check whether an expression
4343 matches a given pattern. This is done by the function
4346 bool ex::match(const ex & pattern);
4347 bool ex::match(const ex & pattern, lst & repls);
4350 This function returns @code{true} when the expression matches the pattern
4351 and @code{false} if it doesn't. If used in the second form, the actual
4352 subexpressions matched by the wildcards get returned in the @code{repls}
4353 object as a list of relations of the form @samp{wildcard == expression}.
4354 If @code{match()} returns false, the state of @code{repls} is undefined.
4355 For reproducible results, the list should be empty when passed to
4356 @code{match()}, but it is also possible to find similarities in multiple
4357 expressions by passing in the result of a previous match.
4359 The matching algorithm works as follows:
4362 @item A single wildcard matches any expression. If one wildcard appears
4363 multiple times in a pattern, it must match the same expression in all
4364 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4365 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4366 @item If the expression is not of the same class as the pattern, the match
4367 fails (i.e. a sum only matches a sum, a function only matches a function,
4369 @item If the pattern is a function, it only matches the same function
4370 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4371 @item Except for sums and products, the match fails if the number of
4372 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4374 @item If there are no subexpressions, the expressions and the pattern must
4375 be equal (in the sense of @code{is_equal()}).
4376 @item Except for sums and products, each subexpression (@code{op()}) must
4377 match the corresponding subexpression of the pattern.
4380 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4381 account for their commutativity and associativity:
4384 @item If the pattern contains a term or factor that is a single wildcard,
4385 this one is used as the @dfn{global wildcard}. If there is more than one
4386 such wildcard, one of them is chosen as the global wildcard in a random
4388 @item Every term/factor of the pattern, except the global wildcard, is
4389 matched against every term of the expression in sequence. If no match is
4390 found, the whole match fails. Terms that did match are not considered in
4392 @item If there are no unmatched terms left, the match succeeds. Otherwise
4393 the match fails unless there is a global wildcard in the pattern, in
4394 which case this wildcard matches the remaining terms.
4397 In general, having more than one single wildcard as a term of a sum or a
4398 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4401 Here are some examples in @command{ginsh} to demonstrate how it works (the
4402 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4403 match fails, and the list of wildcard replacements otherwise):
4406 > match((x+y)^a,(x+y)^a);
4408 > match((x+y)^a,(x+y)^b);
4410 > match((x+y)^a,$1^$2);
4412 > match((x+y)^a,$1^$1);
4414 > match((x+y)^(x+y),$1^$1);
4416 > match((x+y)^(x+y),$1^$2);
4418 > match((a+b)*(a+c),($1+b)*($1+c));
4420 > match((a+b)*(a+c),(a+$1)*(a+$2));
4422 (Unpredictable. The result might also be [$1==c,$2==b].)
4423 > match((a+b)*(a+c),($1+$2)*($1+$3));
4424 (The result is undefined. Due to the sequential nature of the algorithm
4425 and the re-ordering of terms in GiNaC, the match for the first factor
4426 may be @{$1==a,$2==b@} in which case the match for the second factor
4427 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4429 > match(a*(x+y)+a*z+b,a*$1+$2);
4430 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4431 @{$1=x+y,$2=a*z+b@}.)
4432 > match(a+b+c+d+e+f,c);
4434 > match(a+b+c+d+e+f,c+$0);
4436 > match(a+b+c+d+e+f,c+e+$0);
4438 > match(a+b,a+b+$0);
4440 > match(a*b^2,a^$1*b^$2);
4442 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4443 even though a==a^1.)
4444 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4446 > match(atan2(y,x^2),atan2(y,$0));
4450 @subsection Matching parts of expressions
4451 @cindex @code{has()}
4452 A more general way to look for patterns in expressions is provided by the
4456 bool ex::has(const ex & pattern);
4459 This function checks whether a pattern is matched by an expression itself or
4460 by any of its subexpressions.
4462 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4463 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4466 > has(x*sin(x+y+2*a),y);
4468 > has(x*sin(x+y+2*a),x+y);
4470 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4471 has the subexpressions "x", "y" and "2*a".)
4472 > has(x*sin(x+y+2*a),x+y+$1);
4474 (But this is possible.)
4475 > has(x*sin(2*(x+y)+2*a),x+y);
4477 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4478 which "x+y" is not a subexpression.)
4481 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4483 > has(4*x^2-x+3,$1*x);
4485 > has(4*x^2+x+3,$1*x);
4487 (Another possible pitfall. The first expression matches because the term
4488 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4489 contains a linear term you should use the coeff() function instead.)
4492 @cindex @code{find()}
4496 bool ex::find(const ex & pattern, lst & found);
4499 works a bit like @code{has()} but it doesn't stop upon finding the first
4500 match. Instead, it appends all found matches to the specified list. If there
4501 are multiple occurrences of the same expression, it is entered only once to
4502 the list. @code{find()} returns false if no matches were found (in
4503 @command{ginsh}, it returns an empty list):
4506 > find(1+x+x^2+x^3,x);
4508 > find(1+x+x^2+x^3,y);
4510 > find(1+x+x^2+x^3,x^$1);
4512 (Note the absence of "x".)
4513 > expand((sin(x)+sin(y))*(a+b));
4514 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4519 @subsection Substituting expressions
4520 @cindex @code{subs()}
4521 Probably the most useful application of patterns is to use them for
4522 substituting expressions with the @code{subs()} method. Wildcards can be
4523 used in the search patterns as well as in the replacement expressions, where
4524 they get replaced by the expressions matched by them. @code{subs()} doesn't
4525 know anything about algebra; it performs purely syntactic substitutions.
4530 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4532 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4534 > subs((a+b+c)^2,a+b==x);
4536 > subs((a+b+c)^2,a+b+$1==x+$1);
4538 > subs(a+2*b,a+b==x);
4540 > subs(4*x^3-2*x^2+5*x-1,x==a);
4542 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4544 > subs(sin(1+sin(x)),sin($1)==cos($1));
4546 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4550 The last example would be written in C++ in this way:
4554 symbol a("a"), b("b"), x("x"), y("y");
4555 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4556 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4557 cout << e.expand() << endl;
4562 @subsection Algebraic substitutions
4563 Supplying the @code{subs_options::algebraic} option to @code{subs()}
4564 enables smarter, algebraic substitutions in products and powers. If you want
4565 to substitute some factors of a product, you only need to list these factors
4566 in your pattern. Furthermore, if an (integer) power of some expression occurs
4567 in your pattern and in the expression that you want the substitution to occur
4568 in, it can be substituted as many times as possible, without getting negative
4571 An example clarifies it all (hopefully):
4574 cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
4575 subs_options::algebraic) << endl;
4576 // --> (y+x)^6+b^6+a^6
4578 cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
4580 // Powers and products are smart, but addition is just the same.
4582 cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
4585 // As I said: addition is just the same.
4587 cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
4588 // --> x^3*b*a^2+2*b
4590 cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
4592 // --> 2*b+x^3*b^(-1)*a^(-2)
4594 cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
4595 // --> -1-2*a^2+4*a^3+5*a
4597 cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
4598 subs_options::algebraic) << endl;
4599 // --> -1+5*x+4*x^3-2*x^2
4600 // You should not really need this kind of patterns very often now.
4601 // But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
4603 cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
4604 subs_options::algebraic) << endl;
4605 // --> cos(1+cos(x))
4607 cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
4608 .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
4609 subs_options::algebraic)) << endl;
4614 @node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
4615 @c node-name, next, previous, up
4616 @section Applying a Function on Subexpressions
4617 @cindex tree traversal
4618 @cindex @code{map()}
4620 Sometimes you may want to perform an operation on specific parts of an
4621 expression while leaving the general structure of it intact. An example
4622 of this would be a matrix trace operation: the trace of a sum is the sum
4623 of the traces of the individual terms. That is, the trace should @dfn{map}
4624 on the sum, by applying itself to each of the sum's operands. It is possible
4625 to do this manually which usually results in code like this:
4630 if (is_a<matrix>(e))
4631 return ex_to<matrix>(e).trace();
4632 else if (is_a<add>(e)) @{
4634 for (size_t i=0; i<e.nops(); i++)
4635 sum += calc_trace(e.op(i));
4637 @} else if (is_a<mul>)(e)) @{
4645 This is, however, slightly inefficient (if the sum is very large it can take
4646 a long time to add the terms one-by-one), and its applicability is limited to
4647 a rather small class of expressions. If @code{calc_trace()} is called with
4648 a relation or a list as its argument, you will probably want the trace to
4649 be taken on both sides of the relation or of all elements of the list.
4651 GiNaC offers the @code{map()} method to aid in the implementation of such
4655 ex ex::map(map_function & f) const;
4656 ex ex::map(ex (*f)(const ex & e)) const;
4659 In the first (preferred) form, @code{map()} takes a function object that
4660 is subclassed from the @code{map_function} class. In the second form, it
4661 takes a pointer to a function that accepts and returns an expression.
4662 @code{map()} constructs a new expression of the same type, applying the
4663 specified function on all subexpressions (in the sense of @code{op()}),
4666 The use of a function object makes it possible to supply more arguments to
4667 the function that is being mapped, or to keep local state information.
4668 The @code{map_function} class declares a virtual function call operator
4669 that you can overload. Here is a sample implementation of @code{calc_trace()}
4670 that uses @code{map()} in a recursive fashion:
4673 struct calc_trace : public map_function @{
4674 ex operator()(const ex &e)
4676 if (is_a<matrix>(e))
4677 return ex_to<matrix>(e).trace();
4678 else if (is_a<mul>(e)) @{
4681 return e.map(*this);
4686 This function object could then be used like this:
4690 ex M = ... // expression with matrices
4691 calc_trace do_trace;
4692 ex tr = do_trace(M);
4696 Here is another example for you to meditate over. It removes quadratic
4697 terms in a variable from an expanded polynomial:
4700 struct map_rem_quad : public map_function @{
4702 map_rem_quad(const ex & var_) : var(var_) @{@}
4704 ex operator()(const ex & e)
4706 if (is_a<add>(e) || is_a<mul>(e))
4707 return e.map(*this);
4708 else if (is_a<power>(e) &&
4709 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4719 symbol x("x"), y("y");
4722 for (int i=0; i<8; i++)
4723 e += pow(x, i) * pow(y, 8-i) * (i+1);
4725 // -> 4*y^5*x^3+5*y^4*x^4+8*y*x^7+7*y^2*x^6+2*y^7*x+6*y^3*x^5+3*y^6*x^2+y^8
4727 map_rem_quad rem_quad(x);
4728 cout << rem_quad(e) << endl;
4729 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4733 @command{ginsh} offers a slightly different implementation of @code{map()}
4734 that allows applying algebraic functions to operands. The second argument
4735 to @code{map()} is an expression containing the wildcard @samp{$0} which
4736 acts as the placeholder for the operands:
4741 > map(a+2*b,sin($0));
4743 > map(@{a,b,c@},$0^2+$0);
4744 @{a^2+a,b^2+b,c^2+c@}
4747 Note that it is only possible to use algebraic functions in the second
4748 argument. You can not use functions like @samp{diff()}, @samp{op()},
4749 @samp{subs()} etc. because these are evaluated immediately:
4752 > map(@{a,b,c@},diff($0,a));
4754 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4755 to "map(@{a,b,c@},0)".
4759 @node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
4760 @c node-name, next, previous, up
4761 @section Visitors and Tree Traversal
4762 @cindex tree traversal
4763 @cindex @code{visitor} (class)
4764 @cindex @code{accept()}
4765 @cindex @code{visit()}
4766 @cindex @code{traverse()}
4767 @cindex @code{traverse_preorder()}
4768 @cindex @code{traverse_postorder()}
4770 Suppose that you need a function that returns a list of all indices appearing
4771 in an arbitrary expression. The indices can have any dimension, and for
4772 indices with variance you always want the covariant version returned.
4774 You can't use @code{get_free_indices()} because you also want to include
4775 dummy indices in the list, and you can't use @code{find()} as it needs
4776 specific index dimensions (and it would require two passes: one for indices
4777 with variance, one for plain ones).
4779 The obvious solution to this problem is a tree traversal with a type switch,
4780 such as the following:
4783 void gather_indices_helper(const ex & e, lst & l)
4785 if (is_a<varidx>(e)) @{
4786 const varidx & vi = ex_to<varidx>(e);
4787 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4788 @} else if (is_a<idx>(e)) @{
4791 size_t n = e.nops();
4792 for (size_t i = 0; i < n; ++i)
4793 gather_indices_helper(e.op(i), l);
4797 lst gather_indices(const ex & e)
4800 gather_indices_helper(e, l);
4807 This works fine but fans of object-oriented programming will feel
4808 uncomfortable with the type switch. One reason is that there is a possibility
4809 for subtle bugs regarding derived classes. If we had, for example, written
4812 if (is_a<idx>(e)) @{
4814 @} else if (is_a<varidx>(e)) @{
4818 in @code{gather_indices_helper}, the code wouldn't have worked because the
4819 first line "absorbs" all classes derived from @code{idx}, including
4820 @code{varidx}, so the special case for @code{varidx} would never have been
4823 Also, for a large number of classes, a type switch like the above can get
4824 unwieldy and inefficient (it's a linear search, after all).
4825 @code{gather_indices_helper} only checks for two classes, but if you had to
4826 write a function that required a different implementation for nearly
4827 every GiNaC class, the result would be very hard to maintain and extend.
4829 The cleanest approach to the problem would be to add a new virtual function
4830 to GiNaC's class hierarchy. In our example, there would be specializations
4831 for @code{idx} and @code{varidx} while the default implementation in
4832 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4833 impossible to add virtual member functions to existing classes without
4834 changing their source and recompiling everything. GiNaC comes with source,
4835 so you could actually do this, but for a small algorithm like the one
4836 presented this would be impractical.
4838 One solution to this dilemma is the @dfn{Visitor} design pattern,
4839 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4840 variation, described in detail in
4841 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4842 virtual functions to the class hierarchy to implement operations, GiNaC
4843 provides a single "bouncing" method @code{accept()} that takes an instance
4844 of a special @code{visitor} class and redirects execution to the one
4845 @code{visit()} virtual function of the visitor that matches the type of
4846 object that @code{accept()} was being invoked on.
4848 Visitors in GiNaC must derive from the global @code{visitor} class as well
4849 as from the class @code{T::visitor} of each class @code{T} they want to
4850 visit, and implement the member functions @code{void visit(const T &)} for
4856 void ex::accept(visitor & v) const;
4859 will then dispatch to the correct @code{visit()} member function of the
4860 specified visitor @code{v} for the type of GiNaC object at the root of the
4861 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4863 Here is an example of a visitor:
4867 : public visitor, // this is required
4868 public add::visitor, // visit add objects
4869 public numeric::visitor, // visit numeric objects
4870 public basic::visitor // visit basic objects
4872 void visit(const add & x)
4873 @{ cout << "called with an add object" << endl; @}
4875 void visit(const numeric & x)
4876 @{ cout << "called with a numeric object" << endl; @}
4878 void visit(const basic & x)
4879 @{ cout << "called with a basic object" << endl; @}
4883 which can be used as follows:
4894 // prints "called with a numeric object"
4896 // prints "called with an add object"
4898 // prints "called with a basic object"
4902 The @code{visit(const basic &)} method gets called for all objects that are
4903 not @code{numeric} or @code{add} and acts as an (optional) default.
4905 From a conceptual point of view, the @code{visit()} methods of the visitor
4906 behave like a newly added virtual function of the visited hierarchy.
4907 In addition, visitors can store state in member variables, and they can
4908 be extended by deriving a new visitor from an existing one, thus building
4909 hierarchies of visitors.
4911 We can now rewrite our index example from above with a visitor:
4914 class gather_indices_visitor
4915 : public visitor, public idx::visitor, public varidx::visitor
4919 void visit(const idx & i)
4924 void visit(const varidx & vi)
4926 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4930 const lst & get_result() // utility function
4939 What's missing is the tree traversal. We could implement it in
4940 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4943 void ex::traverse_preorder(visitor & v) const;
4944 void ex::traverse_postorder(visitor & v) const;
4945 void ex::traverse(visitor & v) const;
4948 @code{traverse_preorder()} visits a node @emph{before} visiting its
4949 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4950 visiting its subexpressions. @code{traverse()} is a synonym for
4951 @code{traverse_preorder()}.
4953 Here is a new implementation of @code{gather_indices()} that uses the visitor
4954 and @code{traverse()}:
4957 lst gather_indices(const ex & e)
4959 gather_indices_visitor v;
4961 return v.get_result();
4965 Alternatively, you could use pre- or postorder iterators for the tree
4969 lst gather_indices(const ex & e)
4971 gather_indices_visitor v;
4972 for (const_preorder_iterator i = e.preorder_begin();
4973 i != e.preorder_end(); ++i) @{
4976 return v.get_result();
4981 @node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
4982 @c node-name, next, previous, up
4983 @section Polynomial arithmetic
4985 @subsection Expanding and collecting
4986 @cindex @code{expand()}
4987 @cindex @code{collect()}
4988 @cindex @code{collect_common_factors()}
4990 A polynomial in one or more variables has many equivalent
4991 representations. Some useful ones serve a specific purpose. Consider
4992 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
4993 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
4994 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
4995 representations are the recursive ones where one collects for exponents
4996 in one of the three variable. Since the factors are themselves
4997 polynomials in the remaining two variables the procedure can be
4998 repeated. In our example, two possibilities would be @math{(4*y + z)*x
4999 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5002 To bring an expression into expanded form, its method
5005 ex ex::expand(unsigned options = 0);
5008 may be called. In our example above, this corresponds to @math{4*x*y +
5009 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5010 GiNaC is not easy to guess you should be prepared to see different
5011 orderings of terms in such sums!
5013 Another useful representation of multivariate polynomials is as a
5014 univariate polynomial in one of the variables with the coefficients
5015 being polynomials in the remaining variables. The method
5016 @code{collect()} accomplishes this task:
5019 ex ex::collect(const ex & s, bool distributed = false);
5022 The first argument to @code{collect()} can also be a list of objects in which
5023 case the result is either a recursively collected polynomial, or a polynomial
5024 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5025 by the @code{distributed} flag.
5027 Note that the original polynomial needs to be in expanded form (for the
5028 variables concerned) in order for @code{collect()} to be able to find the
5029 coefficients properly.
5031 The following @command{ginsh} transcript shows an application of @code{collect()}
5032 together with @code{find()}:
5035 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5036 d*p*sin(x)+p*sin(x)+q*d*sin(x)+q*sin(y)+d*sin(x)+q*d*sin(y)+sin(y)+d*sin(y)
5037 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5038 > collect(a,@{p,q@});
5039 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5040 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5041 > collect(a,find(a,sin($1)));
5042 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5043 > collect(a,@{find(a,sin($1)),p,q@});
5044 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5045 > collect(a,@{find(a,sin($1)),d@});
5046 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5049 Polynomials can often be brought into a more compact form by collecting
5050 common factors from the terms of sums. This is accomplished by the function
5053 ex collect_common_factors(const ex & e);
5056 This function doesn't perform a full factorization but only looks for
5057 factors which are already explicitly present:
5060 > collect_common_factors(a*x+a*y);
5062 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5064 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5065 (c+a)*a*(x*y+y^2+x)*b
5068 @subsection Degree and coefficients
5069 @cindex @code{degree()}
5070 @cindex @code{ldegree()}
5071 @cindex @code{coeff()}
5073 The degree and low degree of a polynomial can be obtained using the two
5077 int ex::degree(const ex & s);
5078 int ex::ldegree(const ex & s);
5081 which also work reliably on non-expanded input polynomials (they even work
5082 on rational functions, returning the asymptotic degree). By definition, the
5083 degree of zero is zero. To extract a coefficient with a certain power from
5084 an expanded polynomial you use
5087 ex ex::coeff(const ex & s, int n);
5090 You can also obtain the leading and trailing coefficients with the methods
5093 ex ex::lcoeff(const ex & s);
5094 ex ex::tcoeff(const ex & s);
5097 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5100 An application is illustrated in the next example, where a multivariate
5101 polynomial is analyzed:
5105 symbol x("x"), y("y");
5106 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5107 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5108 ex Poly = PolyInp.expand();
5110 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5111 cout << "The x^" << i << "-coefficient is "
5112 << Poly.coeff(x,i) << endl;
5114 cout << "As polynomial in y: "
5115 << Poly.collect(y) << endl;
5119 When run, it returns an output in the following fashion:
5122 The x^0-coefficient is y^2+11*y
5123 The x^1-coefficient is 5*y^2-2*y
5124 The x^2-coefficient is -1
5125 The x^3-coefficient is 4*y
5126 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5129 As always, the exact output may vary between different versions of GiNaC
5130 or even from run to run since the internal canonical ordering is not
5131 within the user's sphere of influence.
5133 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5134 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5135 with non-polynomial expressions as they not only work with symbols but with
5136 constants, functions and indexed objects as well:
5140 symbol a("a"), b("b"), c("c"), x("x");
5141 idx i(symbol("i"), 3);
5143 ex e = pow(sin(x) - cos(x), 4);
5144 cout << e.degree(cos(x)) << endl;
5146 cout << e.expand().coeff(sin(x), 3) << endl;
5149 e = indexed(a+b, i) * indexed(b+c, i);
5150 e = e.expand(expand_options::expand_indexed);
5151 cout << e.collect(indexed(b, i)) << endl;
5152 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5157 @subsection Polynomial division
5158 @cindex polynomial division
5161 @cindex pseudo-remainder
5162 @cindex @code{quo()}
5163 @cindex @code{rem()}
5164 @cindex @code{prem()}
5165 @cindex @code{divide()}
5170 ex quo(const ex & a, const ex & b, const ex & x);
5171 ex rem(const ex & a, const ex & b, const ex & x);
5174 compute the quotient and remainder of univariate polynomials in the variable
5175 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5177 The additional function
5180 ex prem(const ex & a, const ex & b, const ex & x);
5183 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5184 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5186 Exact division of multivariate polynomials is performed by the function
5189 bool divide(const ex & a, const ex & b, ex & q);
5192 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5193 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5194 in which case the value of @code{q} is undefined.
5197 @subsection Unit, content and primitive part
5198 @cindex @code{unit()}
5199 @cindex @code{content()}
5200 @cindex @code{primpart()}
5201 @cindex @code{unitcontprim()}
5206 ex ex::unit(const ex & x);
5207 ex ex::content(const ex & x);
5208 ex ex::primpart(const ex & x);
5209 ex ex::primpart(const ex & x, const ex & c);
5212 return the unit part, content part, and primitive polynomial of a multivariate
5213 polynomial with respect to the variable @samp{x} (the unit part being the sign
5214 of the leading coefficient, the content part being the GCD of the coefficients,
5215 and the primitive polynomial being the input polynomial divided by the unit and
5216 content parts). The second variant of @code{primpart()} expects the previously
5217 calculated content part of the polynomial in @code{c}, which enables it to
5218 work faster in the case where the content part has already been computed. The
5219 product of unit, content, and primitive part is the original polynomial.
5221 Additionally, the method
5224 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5227 computes the unit, content, and primitive parts in one go, returning them
5228 in @code{u}, @code{c}, and @code{p}, respectively.
5231 @subsection GCD, LCM and resultant
5234 @cindex @code{gcd()}
5235 @cindex @code{lcm()}
5237 The functions for polynomial greatest common divisor and least common
5238 multiple have the synopsis
5241 ex gcd(const ex & a, const ex & b);
5242 ex lcm(const ex & a, const ex & b);
5245 The functions @code{gcd()} and @code{lcm()} accept two expressions
5246 @code{a} and @code{b} as arguments and return a new expression, their
5247 greatest common divisor or least common multiple, respectively. If the
5248 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5249 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5250 the coefficients must be rationals.
5253 #include <ginac/ginac.h>
5254 using namespace GiNaC;
5258 symbol x("x"), y("y"), z("z");
5259 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5260 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5262 ex P_gcd = gcd(P_a, P_b);
5264 ex P_lcm = lcm(P_a, P_b);
5265 // 4*x*y^2 + 13*y*x*z + 20*y^3 + 81*y^2*z + 67*y*z^2 + 3*x*z^2 + 12*z^3
5270 @cindex @code{resultant()}
5272 The resultant of two expressions only makes sense with polynomials.
5273 It is always computed with respect to a specific symbol within the
5274 expressions. The function has the interface
5277 ex resultant(const ex & a, const ex & b, const ex & s);
5280 Resultants are symmetric in @code{a} and @code{b}. The following example
5281 computes the resultant of two expressions with respect to @code{x} and
5282 @code{y}, respectively:
5285 #include <ginac/ginac.h>
5286 using namespace GiNaC;
5290 symbol x("x"), y("y");
5292 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5295 r = resultant(e1, e2, x);
5297 r = resultant(e1, e2, y);
5302 @subsection Square-free decomposition
5303 @cindex square-free decomposition
5304 @cindex factorization
5305 @cindex @code{sqrfree()}
5307 GiNaC still lacks proper factorization support. Some form of
5308 factorization is, however, easily implemented by noting that factors
5309 appearing in a polynomial with power two or more also appear in the
5310 derivative and hence can easily be found by computing the GCD of the
5311 original polynomial and its derivatives. Any decent system has an
5312 interface for this so called square-free factorization. So we provide
5315 ex sqrfree(const ex & a, const lst & l = lst());
5317 Here is an example that by the way illustrates how the exact form of the
5318 result may slightly depend on the order of differentiation, calling for
5319 some care with subsequent processing of the result:
5322 symbol x("x"), y("y");
5323 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5325 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5326 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5328 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5329 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5331 cout << sqrfree(BiVarPol) << endl;
5332 // -> depending on luck, any of the above
5335 Note also, how factors with the same exponents are not fully factorized
5339 @node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
5340 @c node-name, next, previous, up
5341 @section Rational expressions
5343 @subsection The @code{normal} method
5344 @cindex @code{normal()}
5345 @cindex simplification
5346 @cindex temporary replacement
5348 Some basic form of simplification of expressions is called for frequently.
5349 GiNaC provides the method @code{.normal()}, which converts a rational function
5350 into an equivalent rational function of the form @samp{numerator/denominator}
5351 where numerator and denominator are coprime. If the input expression is already
5352 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5353 otherwise it performs fraction addition and multiplication.
5355 @code{.normal()} can also be used on expressions which are not rational functions
5356 as it will replace all non-rational objects (like functions or non-integer
5357 powers) by temporary symbols to bring the expression to the domain of rational
5358 functions before performing the normalization, and re-substituting these
5359 symbols afterwards. This algorithm is also available as a separate method
5360 @code{.to_rational()}, described below.
5362 This means that both expressions @code{t1} and @code{t2} are indeed
5363 simplified in this little code snippet:
5368 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5369 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5370 std::cout << "t1 is " << t1.normal() << std::endl;
5371 std::cout << "t2 is " << t2.normal() << std::endl;
5375 Of course this works for multivariate polynomials too, so the ratio of
5376 the sample-polynomials from the section about GCD and LCM above would be
5377 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5380 @subsection Numerator and denominator
5383 @cindex @code{numer()}
5384 @cindex @code{denom()}
5385 @cindex @code{numer_denom()}
5387 The numerator and denominator of an expression can be obtained with
5392 ex ex::numer_denom();
5395 These functions will first normalize the expression as described above and
5396 then return the numerator, denominator, or both as a list, respectively.
5397 If you need both numerator and denominator, calling @code{numer_denom()} is
5398 faster than using @code{numer()} and @code{denom()} separately.
5401 @subsection Converting to a polynomial or rational expression
5402 @cindex @code{to_polynomial()}
5403 @cindex @code{to_rational()}
5405 Some of the methods described so far only work on polynomials or rational
5406 functions. GiNaC provides a way to extend the domain of these functions to
5407 general expressions by using the temporary replacement algorithm described
5408 above. You do this by calling
5411 ex ex::to_polynomial(exmap & m);
5412 ex ex::to_polynomial(lst & l);
5416 ex ex::to_rational(exmap & m);
5417 ex ex::to_rational(lst & l);
5420 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5421 will be filled with the generated temporary symbols and their replacement
5422 expressions in a format that can be used directly for the @code{subs()}
5423 method. It can also already contain a list of replacements from an earlier
5424 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5425 possible to use it on multiple expressions and get consistent results.
5427 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5428 is probably best illustrated with an example:
5432 symbol x("x"), y("y");
5433 ex a = 2*x/sin(x) - y/(3*sin(x));
5437 ex p = a.to_polynomial(lp);
5438 cout << " = " << p << "\n with " << lp << endl;
5439 // = symbol3*symbol2*y+2*symbol2*x
5440 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5443 ex r = a.to_rational(lr);
5444 cout << " = " << r << "\n with " << lr << endl;
5445 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5446 // with @{symbol4==sin(x)@}
5450 The following more useful example will print @samp{sin(x)-cos(x)}:
5455 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5456 ex b = sin(x) + cos(x);
5459 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5460 cout << q.subs(m) << endl;
5465 @node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
5466 @c node-name, next, previous, up
5467 @section Symbolic differentiation
5468 @cindex differentiation
5469 @cindex @code{diff()}
5471 @cindex product rule
5473 GiNaC's objects know how to differentiate themselves. Thus, a
5474 polynomial (class @code{add}) knows that its derivative is the sum of
5475 the derivatives of all the monomials:
5479 symbol x("x"), y("y"), z("z");
5480 ex P = pow(x, 5) + pow(x, 2) + y;
5482 cout << P.diff(x,2) << endl;
5484 cout << P.diff(y) << endl; // 1
5486 cout << P.diff(z) << endl; // 0
5491 If a second integer parameter @var{n} is given, the @code{diff} method
5492 returns the @var{n}th derivative.
5494 If @emph{every} object and every function is told what its derivative
5495 is, all derivatives of composed objects can be calculated using the
5496 chain rule and the product rule. Consider, for instance the expression
5497 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5498 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5499 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5500 out that the composition is the generating function for Euler Numbers,
5501 i.e. the so called @var{n}th Euler number is the coefficient of
5502 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5503 identity to code a function that generates Euler numbers in just three
5506 @cindex Euler numbers
5508 #include <ginac/ginac.h>
5509 using namespace GiNaC;
5511 ex EulerNumber(unsigned n)
5514 const ex generator = pow(cosh(x),-1);
5515 return generator.diff(x,n).subs(x==0);
5520 for (unsigned i=0; i<11; i+=2)
5521 std::cout << EulerNumber(i) << std::endl;
5526 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5527 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5528 @code{i} by two since all odd Euler numbers vanish anyways.
5531 @node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
5532 @c node-name, next, previous, up
5533 @section Series expansion
5534 @cindex @code{series()}
5535 @cindex Taylor expansion
5536 @cindex Laurent expansion
5537 @cindex @code{pseries} (class)
5538 @cindex @code{Order()}
5540 Expressions know how to expand themselves as a Taylor series or (more
5541 generally) a Laurent series. As in most conventional Computer Algebra
5542 Systems, no distinction is made between those two. There is a class of
5543 its own for storing such series (@code{class pseries}) and a built-in
5544 function (called @code{Order}) for storing the order term of the series.
5545 As a consequence, if you want to work with series, i.e. multiply two
5546 series, you need to call the method @code{ex::series} again to convert
5547 it to a series object with the usual structure (expansion plus order
5548 term). A sample application from special relativity could read:
5551 #include <ginac/ginac.h>
5552 using namespace std;
5553 using namespace GiNaC;
5557 symbol v("v"), c("c");
5559 ex gamma = 1/sqrt(1 - pow(v/c,2));
5560 ex mass_nonrel = gamma.series(v==0, 10);
5562 cout << "the relativistic mass increase with v is " << endl
5563 << mass_nonrel << endl;
5565 cout << "the inverse square of this series is " << endl
5566 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5570 Only calling the series method makes the last output simplify to
5571 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5572 series raised to the power @math{-2}.
5574 @cindex Machin's formula
5575 As another instructive application, let us calculate the numerical
5576 value of Archimedes' constant
5580 (for which there already exists the built-in constant @code{Pi})
5581 using John Machin's amazing formula
5583 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5586 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5588 This equation (and similar ones) were used for over 200 years for
5589 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5590 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5591 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5592 order term with it and the question arises what the system is supposed
5593 to do when the fractions are plugged into that order term. The solution
5594 is to use the function @code{series_to_poly()} to simply strip the order
5598 #include <ginac/ginac.h>
5599 using namespace GiNaC;
5601 ex machin_pi(int degr)
5604 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5605 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5606 -4*pi_expansion.subs(x==numeric(1,239));
5612 using std::cout; // just for fun, another way of...
5613 using std::endl; // ...dealing with this namespace std.
5615 for (int i=2; i<12; i+=2) @{
5616 pi_frac = machin_pi(i);
5617 cout << i << ":\t" << pi_frac << endl
5618 << "\t" << pi_frac.evalf() << endl;
5624 Note how we just called @code{.series(x,degr)} instead of
5625 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5626 method @code{series()}: if the first argument is a symbol the expression
5627 is expanded in that symbol around point @code{0}. When you run this
5628 program, it will type out:
5632 3.1832635983263598326
5633 4: 5359397032/1706489875
5634 3.1405970293260603143
5635 6: 38279241713339684/12184551018734375
5636 3.141621029325034425
5637 8: 76528487109180192540976/24359780855939418203125
5638 3.141591772182177295
5639 10: 327853873402258685803048818236/104359128170408663038552734375
5640 3.1415926824043995174
5644 @node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
5645 @c node-name, next, previous, up
5646 @section Symmetrization
5647 @cindex @code{symmetrize()}
5648 @cindex @code{antisymmetrize()}
5649 @cindex @code{symmetrize_cyclic()}
5654 ex ex::symmetrize(const lst & l);
5655 ex ex::antisymmetrize(const lst & l);
5656 ex ex::symmetrize_cyclic(const lst & l);
5659 symmetrize an expression by returning the sum over all symmetric,
5660 antisymmetric or cyclic permutations of the specified list of objects,
5661 weighted by the number of permutations.
5663 The three additional methods
5666 ex ex::symmetrize();
5667 ex ex::antisymmetrize();
5668 ex ex::symmetrize_cyclic();
5671 symmetrize or antisymmetrize an expression over its free indices.
5673 Symmetrization is most useful with indexed expressions but can be used with
5674 almost any kind of object (anything that is @code{subs()}able):
5678 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5679 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5681 cout << indexed(A, i, j).symmetrize() << endl;
5682 // -> 1/2*A.j.i+1/2*A.i.j
5683 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5684 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5685 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5686 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5690 @node Built-in Functions, Multiple polylogarithms, Symmetrization, Methods and Functions
5691 @c node-name, next, previous, up
5692 @section Predefined mathematical functions
5694 @subsection Overview
5696 GiNaC contains the following predefined mathematical functions:
5699 @multitable @columnfractions .30 .70
5700 @item @strong{Name} @tab @strong{Function}
5703 @cindex @code{abs()}
5704 @item @code{csgn(x)}
5706 @cindex @code{conjugate()}
5707 @item @code{conjugate(x)}
5708 @tab complex conjugation
5709 @cindex @code{csgn()}
5710 @item @code{sqrt(x)}
5711 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5712 @cindex @code{sqrt()}
5715 @cindex @code{sin()}
5718 @cindex @code{cos()}
5721 @cindex @code{tan()}
5722 @item @code{asin(x)}
5724 @cindex @code{asin()}
5725 @item @code{acos(x)}
5727 @cindex @code{acos()}
5728 @item @code{atan(x)}
5729 @tab inverse tangent
5730 @cindex @code{atan()}
5731 @item @code{atan2(y, x)}
5732 @tab inverse tangent with two arguments
5733 @item @code{sinh(x)}
5734 @tab hyperbolic sine
5735 @cindex @code{sinh()}
5736 @item @code{cosh(x)}
5737 @tab hyperbolic cosine
5738 @cindex @code{cosh()}
5739 @item @code{tanh(x)}
5740 @tab hyperbolic tangent
5741 @cindex @code{tanh()}
5742 @item @code{asinh(x)}
5743 @tab inverse hyperbolic sine
5744 @cindex @code{asinh()}
5745 @item @code{acosh(x)}
5746 @tab inverse hyperbolic cosine
5747 @cindex @code{acosh()}
5748 @item @code{atanh(x)}
5749 @tab inverse hyperbolic tangent
5750 @cindex @code{atanh()}
5752 @tab exponential function
5753 @cindex @code{exp()}
5755 @tab natural logarithm
5756 @cindex @code{log()}
5759 @cindex @code{Li2()}
5760 @item @code{Li(m, x)}
5761 @tab classical polylogarithm as well as multiple polylogarithm
5763 @item @code{G(a, y)}
5764 @tab multiple polylogarithm
5766 @item @code{G(a, s, y)}
5767 @tab multiple polylogarithm with explicit signs for the imaginary parts
5769 @item @code{S(n, p, x)}
5770 @tab Nielsen's generalized polylogarithm
5772 @item @code{H(m, x)}
5773 @tab harmonic polylogarithm
5775 @item @code{zeta(m)}
5776 @tab Riemann's zeta function as well as multiple zeta value
5777 @cindex @code{zeta()}
5778 @item @code{zeta(m, s)}
5779 @tab alternating Euler sum
5780 @cindex @code{zeta()}
5781 @item @code{zetaderiv(n, x)}
5782 @tab derivatives of Riemann's zeta function
5783 @item @code{tgamma(x)}
5785 @cindex @code{tgamma()}
5786 @cindex gamma function
5787 @item @code{lgamma(x)}
5788 @tab logarithm of gamma function
5789 @cindex @code{lgamma()}
5790 @item @code{beta(x, y)}
5791 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5792 @cindex @code{beta()}
5794 @tab psi (digamma) function
5795 @cindex @code{psi()}
5796 @item @code{psi(n, x)}
5797 @tab derivatives of psi function (polygamma functions)
5798 @item @code{factorial(n)}
5799 @tab factorial function @math{n!}
5800 @cindex @code{factorial()}
5801 @item @code{binomial(n, k)}
5802 @tab binomial coefficients
5803 @cindex @code{binomial()}
5804 @item @code{Order(x)}
5805 @tab order term function in truncated power series
5806 @cindex @code{Order()}
5811 For functions that have a branch cut in the complex plane GiNaC follows
5812 the conventions for C++ as defined in the ANSI standard as far as
5813 possible. In particular: the natural logarithm (@code{log}) and the
5814 square root (@code{sqrt}) both have their branch cuts running along the
5815 negative real axis where the points on the axis itself belong to the
5816 upper part (i.e. continuous with quadrant II). The inverse
5817 trigonometric and hyperbolic functions are not defined for complex
5818 arguments by the C++ standard, however. In GiNaC we follow the
5819 conventions used by CLN, which in turn follow the carefully designed
5820 definitions in the Common Lisp standard. It should be noted that this
5821 convention is identical to the one used by the C99 standard and by most
5822 serious CAS. It is to be expected that future revisions of the C++
5823 standard incorporate these functions in the complex domain in a manner
5824 compatible with C99.
5826 @node Multiple polylogarithms, Complex Conjugation, Built-in Functions, Methods and Functions
5827 @c node-name, next, previous, up
5828 @subsection Multiple polylogarithms
5830 @cindex polylogarithm
5831 @cindex Nielsen's generalized polylogarithm
5832 @cindex harmonic polylogarithm
5833 @cindex multiple zeta value
5834 @cindex alternating Euler sum
5835 @cindex multiple polylogarithm
5837 The multiple polylogarithm is the most generic member of a family of functions,
5838 to which others like the harmonic polylogarithm, Nielsen's generalized
5839 polylogarithm and the multiple zeta value belong.
5840 Everyone of these functions can also be written as a multiple polylogarithm with specific
5841 parameters. This whole family of functions is therefore often referred to simply as
5842 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5843 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5844 @code{Li} and @code{G} in principle represent the same function, the different
5845 notations are more natural to the series representation or the integral
5846 representation, respectively.
5848 To facilitate the discussion of these functions we distinguish between indices and
5849 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5850 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5852 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5853 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5854 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5855 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5856 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5857 @code{s} is not given, the signs default to +1.
5858 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5859 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5860 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5861 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5862 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5864 The functions print in LaTeX format as
5866 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5872 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5875 $\zeta(m_1,m_2,\ldots,m_k)$.
5877 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5878 are printed with a line above, e.g.
5880 $\zeta(5,\overline{2})$.
5882 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5884 Definitions and analytical as well as numerical properties of multiple polylogarithms
5885 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5886 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5887 except for a few differences which will be explicitly stated in the following.
5889 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5890 that the indices and arguments are understood to be in the same order as in which they appear in
5891 the series representation. This means
5893 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5896 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5899 $\zeta(1,2)$ evaluates to infinity.
5901 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5904 The functions only evaluate if the indices are integers greater than zero, except for the indices
5905 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5906 will be interpreted as the sequence of signs for the corresponding indices
5907 @code{m} or the sign of the imaginary part for the
5908 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5909 @code{zeta(lst(3,4), lst(-1,1))} means
5911 $\zeta(\overline{3},4)$
5914 @code{G(lst(a,b), lst(-1,1), c)} means
5916 $G(a-0\epsilon,b+0\epsilon;c)$.
5918 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5919 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5920 e.g. @code{lst(0,0,-1,0,1,0,0)}, @code{lst(0,0,-1,2,0,0)} and @code{lst(-3,2,0,0)} are equivalent as
5921 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5922 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5923 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5924 evaluates also for negative integers and positive even integers. For example:
5927 > Li(@{3,1@},@{x,1@});
5930 -zeta(@{3,2@},@{-1,-1@})
5935 It is easy to tell for a given function into which other function it can be rewritten, may
5936 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5937 with negative indices or trailing zeros (the example above gives a hint). Signs can
5938 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5939 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5940 @code{Li} (@code{eval()} already cares for the possible downgrade):
5943 > convert_H_to_Li(@{0,-2,-1,3@},x);
5944 Li(@{3,1,3@},@{-x,1,-1@})
5945 > convert_H_to_Li(@{2,-1,0@},x);
5946 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5949 Every function can be numerically evaluated for
5950 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5951 global variable @code{Digits}:
5956 > evalf(zeta(@{3,1,3,1@}));
5957 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5960 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5961 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5963 If a function evaluates to infinity, no exceptions are raised, but the function is returned
5968 In long expressions this helps a lot with debugging, because you can easily spot
5969 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
5970 cancellations of divergencies happen.
5972 Useful publications:
5974 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
5975 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
5977 @cite{Harmonic Polylogarithms},
5978 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
5980 @cite{Special Values of Multiple Polylogarithms},
5981 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
5983 @cite{Numerical Evaluation of Multiple Polylogarithms},
5984 J.Vollinga, S.Weinzierl, hep-ph/0410259
5986 @node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
5987 @c node-name, next, previous, up
5988 @section Complex Conjugation
5990 @cindex @code{conjugate()}
5998 returns the complex conjugate of the expression. For all built-in functions and objects the
5999 conjugation gives the expected results:
6003 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6007 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6008 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6009 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6010 // -> -gamma5*gamma~b*gamma~a
6014 For symbols in the complex domain the conjugation can not be evaluated and the GiNaC function
6015 @code{conjugate} is returned. GiNaC functions conjugate by applying the conjugation to their
6016 arguments. This is the default strategy. If you want to define your own functions and want to
6017 change this behavior, you have to supply a specialized conjugation method for your function
6018 (see @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an example).
6020 @node Solving Linear Systems of Equations, Input/Output, Complex Conjugation, Methods and Functions
6021 @c node-name, next, previous, up
6022 @section Solving Linear Systems of Equations
6023 @cindex @code{lsolve()}
6025 The function @code{lsolve()} provides a convenient wrapper around some
6026 matrix operations that comes in handy when a system of linear equations
6030 ex lsolve(const ex & eqns, const ex & symbols,
6031 unsigned options = solve_algo::automatic);
6034 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6035 @code{relational}) while @code{symbols} is a @code{lst} of
6036 indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
6039 It returns the @code{lst} of solutions as an expression. As an example,
6040 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6044 symbol a("a"), b("b"), x("x"), y("y");
6046 eqns = a*x+b*y==3, x-y==b;
6048 cout << lsolve(eqns, vars) << endl;
6049 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6052 When the linear equations @code{eqns} are underdetermined, the solution
6053 will contain one or more tautological entries like @code{x==x},
6054 depending on the rank of the system. When they are overdetermined, the
6055 solution will be an empty @code{lst}. Note the third optional parameter
6056 to @code{lsolve()}: it accepts the same parameters as
6057 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6061 @node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
6062 @c node-name, next, previous, up
6063 @section Input and output of expressions
6066 @subsection Expression output
6068 @cindex output of expressions
6070 Expressions can simply be written to any stream:
6075 ex e = 4.5*I+pow(x,2)*3/2;
6076 cout << e << endl; // prints '4.5*I+3/2*x^2'
6080 The default output format is identical to the @command{ginsh} input syntax and
6081 to that used by most computer algebra systems, but not directly pastable
6082 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6083 is printed as @samp{x^2}).
6085 It is possible to print expressions in a number of different formats with
6086 a set of stream manipulators;
6089 std::ostream & dflt(std::ostream & os);
6090 std::ostream & latex(std::ostream & os);
6091 std::ostream & tree(std::ostream & os);
6092 std::ostream & csrc(std::ostream & os);
6093 std::ostream & csrc_float(std::ostream & os);
6094 std::ostream & csrc_double(std::ostream & os);
6095 std::ostream & csrc_cl_N(std::ostream & os);
6096 std::ostream & index_dimensions(std::ostream & os);
6097 std::ostream & no_index_dimensions(std::ostream & os);
6100 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6101 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6102 @code{print_csrc()} functions, respectively.
6105 All manipulators affect the stream state permanently. To reset the output
6106 format to the default, use the @code{dflt} manipulator:
6110 cout << latex; // all output to cout will be in LaTeX format from
6112 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6113 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6114 cout << dflt; // revert to default output format
6115 cout << e << endl; // prints '4.5*I+3/2*x^2'
6119 If you don't want to affect the format of the stream you're working with,
6120 you can output to a temporary @code{ostringstream} like this:
6125 s << latex << e; // format of cout remains unchanged
6126 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6131 @cindex @code{csrc_float}
6132 @cindex @code{csrc_double}
6133 @cindex @code{csrc_cl_N}
6134 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6135 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6136 format that can be directly used in a C or C++ program. The three possible
6137 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6138 classes provided by the CLN library):
6142 cout << "f = " << csrc_float << e << ";\n";
6143 cout << "d = " << csrc_double << e << ";\n";
6144 cout << "n = " << csrc_cl_N << e << ";\n";
6148 The above example will produce (note the @code{x^2} being converted to
6152 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6153 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6154 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6158 The @code{tree} manipulator allows dumping the internal structure of an
6159 expression for debugging purposes:
6170 add, hash=0x0, flags=0x3, nops=2
6171 power, hash=0x0, flags=0x3, nops=2
6172 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6173 2 (numeric), hash=0x6526b0fa, flags=0xf
6174 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6177 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6181 @cindex @code{latex}
6182 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6183 It is rather similar to the default format but provides some braces needed
6184 by LaTeX for delimiting boxes and also converts some common objects to
6185 conventional LaTeX names. It is possible to give symbols a special name for
6186 LaTeX output by supplying it as a second argument to the @code{symbol}
6189 For example, the code snippet
6193 symbol x("x", "\\circ");
6194 ex e = lgamma(x).series(x==0,3);
6195 cout << latex << e << endl;
6202 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6203 +\mathcal@{O@}(\circ^@{3@})
6206 @cindex @code{index_dimensions}
6207 @cindex @code{no_index_dimensions}
6208 Index dimensions are normally hidden in the output. To make them visible, use
6209 the @code{index_dimensions} manipulator. The dimensions will be written in
6210 square brackets behind each index value in the default and LaTeX output
6215 symbol x("x"), y("y");
6216 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6217 ex e = indexed(x, mu) * indexed(y, nu);
6220 // prints 'x~mu*y~nu'
6221 cout << index_dimensions << e << endl;
6222 // prints 'x~mu[4]*y~nu[4]'
6223 cout << no_index_dimensions << e << endl;
6224 // prints 'x~mu*y~nu'
6229 @cindex Tree traversal
6230 If you need any fancy special output format, e.g. for interfacing GiNaC
6231 with other algebra systems or for producing code for different
6232 programming languages, you can always traverse the expression tree yourself:
6235 static void my_print(const ex & e)
6237 if (is_a<function>(e))
6238 cout << ex_to<function>(e).get_name();
6240 cout << ex_to<basic>(e).class_name();
6242 size_t n = e.nops();
6244 for (size_t i=0; i<n; i++) @{
6256 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6264 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6265 symbol(y))),numeric(-2)))
6268 If you need an output format that makes it possible to accurately
6269 reconstruct an expression by feeding the output to a suitable parser or
6270 object factory, you should consider storing the expression in an
6271 @code{archive} object and reading the object properties from there.
6272 See the section on archiving for more information.
6275 @subsection Expression input
6276 @cindex input of expressions
6278 GiNaC provides no way to directly read an expression from a stream because
6279 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6280 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6281 @code{y} you defined in your program and there is no way to specify the
6282 desired symbols to the @code{>>} stream input operator.
6284 Instead, GiNaC lets you construct an expression from a string, specifying the
6285 list of symbols to be used:
6289 symbol x("x"), y("y");
6290 ex e("2*x+sin(y)", lst(x, y));
6294 The input syntax is the same as that used by @command{ginsh} and the stream
6295 output operator @code{<<}. The symbols in the string are matched by name to
6296 the symbols in the list and if GiNaC encounters a symbol not specified in
6297 the list it will throw an exception.
6299 With this constructor, it's also easy to implement interactive GiNaC programs:
6304 #include <stdexcept>
6305 #include <ginac/ginac.h>
6306 using namespace std;
6307 using namespace GiNaC;
6314 cout << "Enter an expression containing 'x': ";
6319 cout << "The derivative of " << e << " with respect to x is ";
6320 cout << e.diff(x) << ".\n";
6321 @} catch (exception &p) @{
6322 cerr << p.what() << endl;
6328 @subsection Archiving
6329 @cindex @code{archive} (class)
6332 GiNaC allows creating @dfn{archives} of expressions which can be stored
6333 to or retrieved from files. To create an archive, you declare an object
6334 of class @code{archive} and archive expressions in it, giving each
6335 expression a unique name:
6339 using namespace std;
6340 #include <ginac/ginac.h>
6341 using namespace GiNaC;
6345 symbol x("x"), y("y"), z("z");
6347 ex foo = sin(x + 2*y) + 3*z + 41;
6351 a.archive_ex(foo, "foo");
6352 a.archive_ex(bar, "the second one");
6356 The archive can then be written to a file:
6360 ofstream out("foobar.gar");
6366 The file @file{foobar.gar} contains all information that is needed to
6367 reconstruct the expressions @code{foo} and @code{bar}.
6369 @cindex @command{viewgar}
6370 The tool @command{viewgar} that comes with GiNaC can be used to view
6371 the contents of GiNaC archive files:
6374 $ viewgar foobar.gar
6375 foo = 41+sin(x+2*y)+3*z
6376 the second one = 42+sin(x+2*y)+3*z
6379 The point of writing archive files is of course that they can later be
6385 ifstream in("foobar.gar");
6390 And the stored expressions can be retrieved by their name:
6397 ex ex1 = a2.unarchive_ex(syms, "foo");
6398 ex ex2 = a2.unarchive_ex(syms, "the second one");
6400 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6401 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6402 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6406 Note that you have to supply a list of the symbols which are to be inserted
6407 in the expressions. Symbols in archives are stored by their name only and
6408 if you don't specify which symbols you have, unarchiving the expression will
6409 create new symbols with that name. E.g. if you hadn't included @code{x} in
6410 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6411 have had no effect because the @code{x} in @code{ex1} would have been a
6412 different symbol than the @code{x} which was defined at the beginning of
6413 the program, although both would appear as @samp{x} when printed.
6415 You can also use the information stored in an @code{archive} object to
6416 output expressions in a format suitable for exact reconstruction. The
6417 @code{archive} and @code{archive_node} classes have a couple of member
6418 functions that let you access the stored properties:
6421 static void my_print2(const archive_node & n)
6424 n.find_string("class", class_name);
6425 cout << class_name << "(";
6427 archive_node::propinfovector p;
6428 n.get_properties(p);
6430 size_t num = p.size();
6431 for (size_t i=0; i<num; i++) @{
6432 const string &name = p[i].name;
6433 if (name == "class")
6435 cout << name << "=";
6437 unsigned count = p[i].count;
6441 for (unsigned j=0; j<count; j++) @{
6442 switch (p[i].type) @{
6443 case archive_node::PTYPE_BOOL: @{
6445 n.find_bool(name, x, j);
6446 cout << (x ? "true" : "false");
6449 case archive_node::PTYPE_UNSIGNED: @{
6451 n.find_unsigned(name, x, j);
6455 case archive_node::PTYPE_STRING: @{
6457 n.find_string(name, x, j);
6458 cout << '\"' << x << '\"';
6461 case archive_node::PTYPE_NODE: @{
6462 const archive_node &x = n.find_ex_node(name, j);
6484 ex e = pow(2, x) - y;
6486 my_print2(ar.get_top_node(0)); cout << endl;
6494 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6495 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6496 overall_coeff=numeric(number="0"))
6499 Be warned, however, that the set of properties and their meaning for each
6500 class may change between GiNaC versions.
6503 @node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
6504 @c node-name, next, previous, up
6505 @chapter Extending GiNaC
6507 By reading so far you should have gotten a fairly good understanding of
6508 GiNaC's design patterns. From here on you should start reading the
6509 sources. All we can do now is issue some recommendations how to tackle
6510 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6511 develop some useful extension please don't hesitate to contact the GiNaC
6512 authors---they will happily incorporate them into future versions.
6515 * What does not belong into GiNaC:: What to avoid.
6516 * Symbolic functions:: Implementing symbolic functions.
6517 * Printing:: Adding new output formats.
6518 * Structures:: Defining new algebraic classes (the easy way).
6519 * Adding classes:: Defining new algebraic classes (the hard way).
6523 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6524 @c node-name, next, previous, up
6525 @section What doesn't belong into GiNaC
6527 @cindex @command{ginsh}
6528 First of all, GiNaC's name must be read literally. It is designed to be
6529 a library for use within C++. The tiny @command{ginsh} accompanying
6530 GiNaC makes this even more clear: it doesn't even attempt to provide a
6531 language. There are no loops or conditional expressions in
6532 @command{ginsh}, it is merely a window into the library for the
6533 programmer to test stuff (or to show off). Still, the design of a
6534 complete CAS with a language of its own, graphical capabilities and all
6535 this on top of GiNaC is possible and is without doubt a nice project for
6538 There are many built-in functions in GiNaC that do not know how to
6539 evaluate themselves numerically to a precision declared at runtime
6540 (using @code{Digits}). Some may be evaluated at certain points, but not
6541 generally. This ought to be fixed. However, doing numerical
6542 computations with GiNaC's quite abstract classes is doomed to be
6543 inefficient. For this purpose, the underlying foundation classes
6544 provided by CLN are much better suited.
6547 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6548 @c node-name, next, previous, up
6549 @section Symbolic functions
6551 The easiest and most instructive way to start extending GiNaC is probably to
6552 create your own symbolic functions. These are implemented with the help of
6553 two preprocessor macros:
6555 @cindex @code{DECLARE_FUNCTION}
6556 @cindex @code{REGISTER_FUNCTION}
6558 DECLARE_FUNCTION_<n>P(<name>)
6559 REGISTER_FUNCTION(<name>, <options>)
6562 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6563 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6564 parameters of type @code{ex} and returns a newly constructed GiNaC
6565 @code{function} object that represents your function.
6567 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6568 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6569 set of options that associate the symbolic function with C++ functions you
6570 provide to implement the various methods such as evaluation, derivative,
6571 series expansion etc. They also describe additional attributes the function
6572 might have, such as symmetry and commutation properties, and a name for
6573 LaTeX output. Multiple options are separated by the member access operator
6574 @samp{.} and can be given in an arbitrary order.
6576 (By the way: in case you are worrying about all the macros above we can
6577 assure you that functions are GiNaC's most macro-intense classes. We have
6578 done our best to avoid macros where we can.)
6580 @subsection A minimal example
6582 Here is an example for the implementation of a function with two arguments
6583 that is not further evaluated:
6586 DECLARE_FUNCTION_2P(myfcn)
6588 REGISTER_FUNCTION(myfcn, dummy())
6591 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6592 in algebraic expressions:
6598 ex e = 2*myfcn(42, 1+3*x) - x;
6600 // prints '2*myfcn(42,1+3*x)-x'
6605 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6606 "no options". A function with no options specified merely acts as a kind of
6607 container for its arguments. It is a pure "dummy" function with no associated
6608 logic (which is, however, sometimes perfectly sufficient).
6610 Let's now have a look at the implementation of GiNaC's cosine function for an
6611 example of how to make an "intelligent" function.
6613 @subsection The cosine function
6615 The GiNaC header file @file{inifcns.h} contains the line
6618 DECLARE_FUNCTION_1P(cos)
6621 which declares to all programs using GiNaC that there is a function @samp{cos}
6622 that takes one @code{ex} as an argument. This is all they need to know to use
6623 this function in expressions.
6625 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6626 is its @code{REGISTER_FUNCTION} line:
6629 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6630 evalf_func(cos_evalf).
6631 derivative_func(cos_deriv).
6632 latex_name("\\cos"));
6635 There are four options defined for the cosine function. One of them
6636 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6637 other three indicate the C++ functions in which the "brains" of the cosine
6638 function are defined.
6640 @cindex @code{hold()}
6642 The @code{eval_func()} option specifies the C++ function that implements
6643 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6644 the same number of arguments as the associated symbolic function (one in this
6645 case) and returns the (possibly transformed or in some way simplified)
6646 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6647 of the automatic evaluation process). If no (further) evaluation is to take
6648 place, the @code{eval_func()} function must return the original function
6649 with @code{.hold()}, to avoid a potential infinite recursion. If your
6650 symbolic functions produce a segmentation fault or stack overflow when
6651 using them in expressions, you are probably missing a @code{.hold()}
6654 The @code{eval_func()} function for the cosine looks something like this
6655 (actually, it doesn't look like this at all, but it should give you an idea
6659 static ex cos_eval(const ex & x)
6661 if ("x is a multiple of 2*Pi")
6663 else if ("x is a multiple of Pi")
6665 else if ("x is a multiple of Pi/2")
6669 else if ("x has the form 'acos(y)'")
6671 else if ("x has the form 'asin(y)'")
6676 return cos(x).hold();
6680 This function is called every time the cosine is used in a symbolic expression:
6686 // this calls cos_eval(Pi), and inserts its return value into
6687 // the actual expression
6694 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6695 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6696 symbolic transformation can be done, the unmodified function is returned
6697 with @code{.hold()}.
6699 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6700 The user has to call @code{evalf()} for that. This is implemented in a
6704 static ex cos_evalf(const ex & x)
6706 if (is_a<numeric>(x))
6707 return cos(ex_to<numeric>(x));
6709 return cos(x).hold();
6713 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6714 in this case the @code{cos()} function for @code{numeric} objects, which in
6715 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6716 isn't really needed here, but reminds us that the corresponding @code{eval()}
6717 function would require it in this place.
6719 Differentiation will surely turn up and so we need to tell @code{cos}
6720 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6721 instance, are then handled automatically by @code{basic::diff} and
6725 static ex cos_deriv(const ex & x, unsigned diff_param)
6731 @cindex product rule
6732 The second parameter is obligatory but uninteresting at this point. It
6733 specifies which parameter to differentiate in a partial derivative in
6734 case the function has more than one parameter, and its main application
6735 is for correct handling of the chain rule.
6737 An implementation of the series expansion is not needed for @code{cos()} as
6738 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6739 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6740 the other hand, does have poles and may need to do Laurent expansion:
6743 static ex tan_series(const ex & x, const relational & rel,
6744 int order, unsigned options)
6746 // Find the actual expansion point
6747 const ex x_pt = x.subs(rel);
6749 if ("x_pt is not an odd multiple of Pi/2")
6750 throw do_taylor(); // tell function::series() to do Taylor expansion
6752 // On a pole, expand sin()/cos()
6753 return (sin(x)/cos(x)).series(rel, order+2, options);
6757 The @code{series()} implementation of a function @emph{must} return a
6758 @code{pseries} object, otherwise your code will crash.
6760 @subsection Function options
6762 GiNaC functions understand several more options which are always
6763 specified as @code{.option(params)}. None of them are required, but you
6764 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6765 is a do-nothing option called @code{dummy()} which you can use to define
6766 functions without any special options.
6769 eval_func(<C++ function>)
6770 evalf_func(<C++ function>)
6771 derivative_func(<C++ function>)
6772 series_func(<C++ function>)
6773 conjugate_func(<C++ function>)
6776 These specify the C++ functions that implement symbolic evaluation,
6777 numeric evaluation, partial derivatives, and series expansion, respectively.
6778 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6779 @code{diff()} and @code{series()}.
6781 The @code{eval_func()} function needs to use @code{.hold()} if no further
6782 automatic evaluation is desired or possible.
6784 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6785 expansion, which is correct if there are no poles involved. If the function
6786 has poles in the complex plane, the @code{series_func()} needs to check
6787 whether the expansion point is on a pole and fall back to Taylor expansion
6788 if it isn't. Otherwise, the pole usually needs to be regularized by some
6789 suitable transformation.
6792 latex_name(const string & n)
6795 specifies the LaTeX code that represents the name of the function in LaTeX
6796 output. The default is to put the function name in an @code{\mbox@{@}}.
6799 do_not_evalf_params()
6802 This tells @code{evalf()} to not recursively evaluate the parameters of the
6803 function before calling the @code{evalf_func()}.
6806 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6809 This allows you to explicitly specify the commutation properties of the
6810 function (@xref{Non-commutative objects}, for an explanation of
6811 (non)commutativity in GiNaC). For example, you can use
6812 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6813 GiNaC treat your function like a matrix. By default, functions inherit the
6814 commutation properties of their first argument.
6817 set_symmetry(const symmetry & s)
6820 specifies the symmetry properties of the function with respect to its
6821 arguments. @xref{Indexed objects}, for an explanation of symmetry
6822 specifications. GiNaC will automatically rearrange the arguments of
6823 symmetric functions into a canonical order.
6825 Sometimes you may want to have finer control over how functions are
6826 displayed in the output. For example, the @code{abs()} function prints
6827 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6828 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6832 print_func<C>(<C++ function>)
6835 option which is explained in the next section.
6837 @subsection Functions with a variable number of arguments
6839 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6840 functions with a fixed number of arguments. Sometimes, though, you may need
6841 to have a function that accepts a variable number of expressions. One way to
6842 accomplish this is to pass variable-length lists as arguments. The
6843 @code{Li()} function uses this method for multiple polylogarithms.
6845 It is also possible to define functions that accept a different number of
6846 parameters under the same function name, such as the @code{psi()} function
6847 which can be called either as @code{psi(z)} (the digamma function) or as
6848 @code{psi(n, z)} (polygamma functions). These are actually two different
6849 functions in GiNaC that, however, have the same name. Defining such
6850 functions is not possible with the macros but requires manually fiddling
6851 with GiNaC internals. If you are interested, please consult the GiNaC source
6852 code for the @code{psi()} function (@file{inifcns.h} and
6853 @file{inifcns_gamma.cpp}).
6856 @node Printing, Structures, Symbolic functions, Extending GiNaC
6857 @c node-name, next, previous, up
6858 @section GiNaC's expression output system
6860 GiNaC allows the output of expressions in a variety of different formats
6861 (@pxref{Input/Output}). This section will explain how expression output
6862 is implemented internally, and how to define your own output formats or
6863 change the output format of built-in algebraic objects. You will also want
6864 to read this section if you plan to write your own algebraic classes or
6867 @cindex @code{print_context} (class)
6868 @cindex @code{print_dflt} (class)
6869 @cindex @code{print_latex} (class)
6870 @cindex @code{print_tree} (class)
6871 @cindex @code{print_csrc} (class)
6872 All the different output formats are represented by a hierarchy of classes
6873 rooted in the @code{print_context} class, defined in the @file{print.h}
6878 the default output format
6880 output in LaTeX mathematical mode
6882 a dump of the internal expression structure (for debugging)
6884 the base class for C source output
6885 @item print_csrc_float
6886 C source output using the @code{float} type
6887 @item print_csrc_double
6888 C source output using the @code{double} type
6889 @item print_csrc_cl_N
6890 C source output using CLN types
6893 The @code{print_context} base class provides two public data members:
6905 @code{s} is a reference to the stream to output to, while @code{options}
6906 holds flags and modifiers. Currently, there is only one flag defined:
6907 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6908 to print the index dimension which is normally hidden.
6910 When you write something like @code{std::cout << e}, where @code{e} is
6911 an object of class @code{ex}, GiNaC will construct an appropriate
6912 @code{print_context} object (of a class depending on the selected output
6913 format), fill in the @code{s} and @code{options} members, and call
6915 @cindex @code{print()}
6917 void ex::print(const print_context & c, unsigned level = 0) const;
6920 which in turn forwards the call to the @code{print()} method of the
6921 top-level algebraic object contained in the expression.
6923 Unlike other methods, GiNaC classes don't usually override their
6924 @code{print()} method to implement expression output. Instead, the default
6925 implementation @code{basic::print(c, level)} performs a run-time double
6926 dispatch to a function selected by the dynamic type of the object and the
6927 passed @code{print_context}. To this end, GiNaC maintains a separate method
6928 table for each class, similar to the virtual function table used for ordinary
6929 (single) virtual function dispatch.
6931 The method table contains one slot for each possible @code{print_context}
6932 type, indexed by the (internally assigned) serial number of the type. Slots
6933 may be empty, in which case GiNaC will retry the method lookup with the
6934 @code{print_context} object's parent class, possibly repeating the process
6935 until it reaches the @code{print_context} base class. If there's still no
6936 method defined, the method table of the algebraic object's parent class
6937 is consulted, and so on, until a matching method is found (eventually it
6938 will reach the combination @code{basic/print_context}, which prints the
6939 object's class name enclosed in square brackets).
6941 You can think of the print methods of all the different classes and output
6942 formats as being arranged in a two-dimensional matrix with one axis listing
6943 the algebraic classes and the other axis listing the @code{print_context}
6946 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6947 to implement printing, but then they won't get any of the benefits of the
6948 double dispatch mechanism (such as the ability for derived classes to
6949 inherit only certain print methods from its parent, or the replacement of
6950 methods at run-time).
6952 @subsection Print methods for classes
6954 The method table for a class is set up either in the definition of the class,
6955 by passing the appropriate @code{print_func<C>()} option to
6956 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
6957 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
6958 can also be used to override existing methods dynamically.
6960 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
6961 be a member function of the class (or one of its parent classes), a static
6962 member function, or an ordinary (global) C++ function. The @code{C} template
6963 parameter specifies the appropriate @code{print_context} type for which the
6964 method should be invoked, while, in the case of @code{set_print_func<>()}, the
6965 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
6966 the class is the one being implemented by
6967 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
6969 For print methods that are member functions, their first argument must be of
6970 a type convertible to a @code{const C &}, and the second argument must be an
6973 For static members and global functions, the first argument must be of a type
6974 convertible to a @code{const T &}, the second argument must be of a type
6975 convertible to a @code{const C &}, and the third argument must be an
6976 @code{unsigned}. A global function will, of course, not have access to
6977 private and protected members of @code{T}.
6979 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
6980 and @code{basic::print()}) is used for proper parenthesizing of the output
6981 (and by @code{print_tree} for proper indentation). It can be used for similar
6982 purposes if you write your own output formats.
6984 The explanations given above may seem complicated, but in practice it's
6985 really simple, as shown in the following example. Suppose that we want to
6986 display exponents in LaTeX output not as superscripts but with little
6987 upwards-pointing arrows. This can be achieved in the following way:
6990 void my_print_power_as_latex(const power & p,
6991 const print_latex & c,
6994 // get the precedence of the 'power' class
6995 unsigned power_prec = p.precedence();
6997 // if the parent operator has the same or a higher precedence
6998 // we need parentheses around the power
6999 if (level >= power_prec)
7002 // print the basis and exponent, each enclosed in braces, and
7003 // separated by an uparrow
7005 p.op(0).print(c, power_prec);
7006 c.s << "@}\\uparrow@{";
7007 p.op(1).print(c, power_prec);
7010 // don't forget the closing parenthesis
7011 if (level >= power_prec)
7017 // a sample expression
7018 symbol x("x"), y("y");
7019 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7021 // switch to LaTeX mode
7024 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7027 // now we replace the method for the LaTeX output of powers with
7029 set_print_func<power, print_latex>(my_print_power_as_latex);
7031 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7042 The first argument of @code{my_print_power_as_latex} could also have been
7043 a @code{const basic &}, the second one a @code{const print_context &}.
7046 The above code depends on @code{mul} objects converting their operands to
7047 @code{power} objects for the purpose of printing.
7050 The output of products including negative powers as fractions is also
7051 controlled by the @code{mul} class.
7054 The @code{power/print_latex} method provided by GiNaC prints square roots
7055 using @code{\sqrt}, but the above code doesn't.
7059 It's not possible to restore a method table entry to its previous or default
7060 value. Once you have called @code{set_print_func()}, you can only override
7061 it with another call to @code{set_print_func()}, but you can't easily go back
7062 to the default behavior again (you can, of course, dig around in the GiNaC
7063 sources, find the method that is installed at startup
7064 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7065 one; that is, after you circumvent the C++ member access control@dots{}).
7067 @subsection Print methods for functions
7069 Symbolic functions employ a print method dispatch mechanism similar to the
7070 one used for classes. The methods are specified with @code{print_func<C>()}
7071 function options. If you don't specify any special print methods, the function
7072 will be printed with its name (or LaTeX name, if supplied), followed by a
7073 comma-separated list of arguments enclosed in parentheses.
7075 For example, this is what GiNaC's @samp{abs()} function is defined like:
7078 static ex abs_eval(const ex & arg) @{ ... @}
7079 static ex abs_evalf(const ex & arg) @{ ... @}
7081 static void abs_print_latex(const ex & arg, const print_context & c)
7083 c.s << "@{|"; arg.print(c); c.s << "|@}";
7086 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7088 c.s << "fabs("; arg.print(c); c.s << ")";
7091 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7092 evalf_func(abs_evalf).
7093 print_func<print_latex>(abs_print_latex).
7094 print_func<print_csrc_float>(abs_print_csrc_float).
7095 print_func<print_csrc_double>(abs_print_csrc_float));
7098 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7099 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7101 There is currently no equivalent of @code{set_print_func()} for functions.
7103 @subsection Adding new output formats
7105 Creating a new output format involves subclassing @code{print_context},
7106 which is somewhat similar to adding a new algebraic class
7107 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7108 that needs to go into the class definition, and a corresponding macro
7109 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7110 Every @code{print_context} class needs to provide a default constructor
7111 and a constructor from an @code{std::ostream} and an @code{unsigned}
7114 Here is an example for a user-defined @code{print_context} class:
7117 class print_myformat : public print_dflt
7119 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7121 print_myformat(std::ostream & os, unsigned opt = 0)
7122 : print_dflt(os, opt) @{@}
7125 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7127 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7130 That's all there is to it. None of the actual expression output logic is
7131 implemented in this class. It merely serves as a selector for choosing
7132 a particular format. The algorithms for printing expressions in the new
7133 format are implemented as print methods, as described above.
7135 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7136 exactly like GiNaC's default output format:
7141 ex e = pow(x, 2) + 1;
7143 // this prints "1+x^2"
7146 // this also prints "1+x^2"
7147 e.print(print_myformat()); cout << endl;
7153 To fill @code{print_myformat} with life, we need to supply appropriate
7154 print methods with @code{set_print_func()}, like this:
7157 // This prints powers with '**' instead of '^'. See the LaTeX output
7158 // example above for explanations.
7159 void print_power_as_myformat(const power & p,
7160 const print_myformat & c,
7163 unsigned power_prec = p.precedence();
7164 if (level >= power_prec)
7166 p.op(0).print(c, power_prec);
7168 p.op(1).print(c, power_prec);
7169 if (level >= power_prec)
7175 // install a new print method for power objects
7176 set_print_func<power, print_myformat>(print_power_as_myformat);
7178 // now this prints "1+x**2"
7179 e.print(print_myformat()); cout << endl;
7181 // but the default format is still "1+x^2"
7187 @node Structures, Adding classes, Printing, Extending GiNaC
7188 @c node-name, next, previous, up
7191 If you are doing some very specialized things with GiNaC, or if you just
7192 need some more organized way to store data in your expressions instead of
7193 anonymous lists, you may want to implement your own algebraic classes.
7194 ('algebraic class' means any class directly or indirectly derived from
7195 @code{basic} that can be used in GiNaC expressions).
7197 GiNaC offers two ways of accomplishing this: either by using the
7198 @code{structure<T>} template class, or by rolling your own class from
7199 scratch. This section will discuss the @code{structure<T>} template which
7200 is easier to use but more limited, while the implementation of custom
7201 GiNaC classes is the topic of the next section. However, you may want to
7202 read both sections because many common concepts and member functions are
7203 shared by both concepts, and it will also allow you to decide which approach
7204 is most suited to your needs.
7206 The @code{structure<T>} template, defined in the GiNaC header file
7207 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7208 or @code{class}) into a GiNaC object that can be used in expressions.
7210 @subsection Example: scalar products
7212 Let's suppose that we need a way to handle some kind of abstract scalar
7213 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7214 product class have to store their left and right operands, which can in turn
7215 be arbitrary expressions. Here is a possible way to represent such a
7216 product in a C++ @code{struct}:
7220 using namespace std;
7222 #include <ginac/ginac.h>
7223 using namespace GiNaC;
7229 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7233 The default constructor is required. Now, to make a GiNaC class out of this
7234 data structure, we need only one line:
7237 typedef structure<sprod_s> sprod;
7240 That's it. This line constructs an algebraic class @code{sprod} which
7241 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7242 expressions like any other GiNaC class:
7246 symbol a("a"), b("b");
7247 ex e = sprod(sprod_s(a, b));
7251 Note the difference between @code{sprod} which is the algebraic class, and
7252 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7253 and @code{right} data members. As shown above, an @code{sprod} can be
7254 constructed from an @code{sprod_s} object.
7256 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7257 you could define a little wrapper function like this:
7260 inline ex make_sprod(ex left, ex right)
7262 return sprod(sprod_s(left, right));
7266 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7267 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7268 @code{get_struct()}:
7272 cout << ex_to<sprod>(e)->left << endl;
7274 cout << ex_to<sprod>(e).get_struct().right << endl;
7279 You only have read access to the members of @code{sprod_s}.
7281 The type definition of @code{sprod} is enough to write your own algorithms
7282 that deal with scalar products, for example:
7287 if (is_a<sprod>(p)) @{
7288 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7289 return make_sprod(sp.right, sp.left);
7300 @subsection Structure output
7302 While the @code{sprod} type is useable it still leaves something to be
7303 desired, most notably proper output:
7308 // -> [structure object]
7312 By default, any structure types you define will be printed as
7313 @samp{[structure object]}. To override this you can either specialize the
7314 template's @code{print()} member function, or specify print methods with
7315 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7316 it's not possible to supply class options like @code{print_func<>()} to
7317 structures, so for a self-contained structure type you need to resort to
7318 overriding the @code{print()} function, which is also what we will do here.
7320 The member functions of GiNaC classes are described in more detail in the
7321 next section, but it shouldn't be hard to figure out what's going on here:
7324 void sprod::print(const print_context & c, unsigned level) const
7326 // tree debug output handled by superclass
7327 if (is_a<print_tree>(c))
7328 inherited::print(c, level);
7330 // get the contained sprod_s object
7331 const sprod_s & sp = get_struct();
7333 // print_context::s is a reference to an ostream
7334 c.s << "<" << sp.left << "|" << sp.right << ">";
7338 Now we can print expressions containing scalar products:
7344 cout << swap_sprod(e) << endl;
7349 @subsection Comparing structures
7351 The @code{sprod} class defined so far still has one important drawback: all
7352 scalar products are treated as being equal because GiNaC doesn't know how to
7353 compare objects of type @code{sprod_s}. This can lead to some confusing
7354 and undesired behavior:
7358 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7360 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7361 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7365 To remedy this, we first need to define the operators @code{==} and @code{<}
7366 for objects of type @code{sprod_s}:
7369 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7371 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7374 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7376 return lhs.left.compare(rhs.left) < 0
7377 ? true : lhs.right.compare(rhs.right) < 0;
7381 The ordering established by the @code{<} operator doesn't have to make any
7382 algebraic sense, but it needs to be well defined. Note that we can't use
7383 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7384 in the implementation of these operators because they would construct
7385 GiNaC @code{relational} objects which in the case of @code{<} do not
7386 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7387 decide which one is algebraically 'less').
7389 Next, we need to change our definition of the @code{sprod} type to let
7390 GiNaC know that an ordering relation exists for the embedded objects:
7393 typedef structure<sprod_s, compare_std_less> sprod;
7396 @code{sprod} objects then behave as expected:
7400 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7401 // -> <a|b>-<a^2|b^2>
7402 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7403 // -> <a|b>+<a^2|b^2>
7404 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7406 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7411 The @code{compare_std_less} policy parameter tells GiNaC to use the
7412 @code{std::less} and @code{std::equal_to} functors to compare objects of
7413 type @code{sprod_s}. By default, these functors forward their work to the
7414 standard @code{<} and @code{==} operators, which we have overloaded.
7415 Alternatively, we could have specialized @code{std::less} and
7416 @code{std::equal_to} for class @code{sprod_s}.
7418 GiNaC provides two other comparison policies for @code{structure<T>}
7419 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7420 which does a bit-wise comparison of the contained @code{T} objects.
7421 This should be used with extreme care because it only works reliably with
7422 built-in integral types, and it also compares any padding (filler bytes of
7423 undefined value) that the @code{T} class might have.
7425 @subsection Subexpressions
7427 Our scalar product class has two subexpressions: the left and right
7428 operands. It might be a good idea to make them accessible via the standard
7429 @code{nops()} and @code{op()} methods:
7432 size_t sprod::nops() const
7437 ex sprod::op(size_t i) const
7441 return get_struct().left;
7443 return get_struct().right;
7445 throw std::range_error("sprod::op(): no such operand");
7450 Implementing @code{nops()} and @code{op()} for container types such as
7451 @code{sprod} has two other nice side effects:
7455 @code{has()} works as expected
7457 GiNaC generates better hash keys for the objects (the default implementation
7458 of @code{calchash()} takes subexpressions into account)
7461 @cindex @code{let_op()}
7462 There is a non-const variant of @code{op()} called @code{let_op()} that
7463 allows replacing subexpressions:
7466 ex & sprod::let_op(size_t i)
7468 // every non-const member function must call this
7469 ensure_if_modifiable();
7473 return get_struct().left;
7475 return get_struct().right;
7477 throw std::range_error("sprod::let_op(): no such operand");
7482 Once we have provided @code{let_op()} we also get @code{subs()} and
7483 @code{map()} for free. In fact, every container class that returns a non-null
7484 @code{nops()} value must either implement @code{let_op()} or provide custom
7485 implementations of @code{subs()} and @code{map()}.
7487 In turn, the availability of @code{map()} enables the recursive behavior of a
7488 couple of other default method implementations, in particular @code{evalf()},
7489 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7490 we probably want to provide our own version of @code{expand()} for scalar
7491 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7492 This is left as an exercise for the reader.
7494 The @code{structure<T>} template defines many more member functions that
7495 you can override by specialization to customize the behavior of your
7496 structures. You are referred to the next section for a description of
7497 some of these (especially @code{eval()}). There is, however, one topic
7498 that shall be addressed here, as it demonstrates one peculiarity of the
7499 @code{structure<T>} template: archiving.
7501 @subsection Archiving structures
7503 If you don't know how the archiving of GiNaC objects is implemented, you
7504 should first read the next section and then come back here. You're back?
7507 To implement archiving for structures it is not enough to provide
7508 specializations for the @code{archive()} member function and the
7509 unarchiving constructor (the @code{unarchive()} function has a default
7510 implementation). You also need to provide a unique name (as a string literal)
7511 for each structure type you define. This is because in GiNaC archives,
7512 the class of an object is stored as a string, the class name.
7514 By default, this class name (as returned by the @code{class_name()} member
7515 function) is @samp{structure} for all structure classes. This works as long
7516 as you have only defined one structure type, but if you use two or more you
7517 need to provide a different name for each by specializing the
7518 @code{get_class_name()} member function. Here is a sample implementation
7519 for enabling archiving of the scalar product type defined above:
7522 const char *sprod::get_class_name() @{ return "sprod"; @}
7524 void sprod::archive(archive_node & n) const
7526 inherited::archive(n);
7527 n.add_ex("left", get_struct().left);
7528 n.add_ex("right", get_struct().right);
7531 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7533 n.find_ex("left", get_struct().left, sym_lst);
7534 n.find_ex("right", get_struct().right, sym_lst);
7538 Note that the unarchiving constructor is @code{sprod::structure} and not
7539 @code{sprod::sprod}, and that we don't need to supply an
7540 @code{sprod::unarchive()} function.
7543 @node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
7544 @c node-name, next, previous, up
7545 @section Adding classes
7547 The @code{structure<T>} template provides an way to extend GiNaC with custom
7548 algebraic classes that is easy to use but has its limitations, the most
7549 severe of which being that you can't add any new member functions to
7550 structures. To be able to do this, you need to write a new class definition
7553 This section will explain how to implement new algebraic classes in GiNaC by
7554 giving the example of a simple 'string' class. After reading this section
7555 you will know how to properly declare a GiNaC class and what the minimum
7556 required member functions are that you have to implement. We only cover the
7557 implementation of a 'leaf' class here (i.e. one that doesn't contain
7558 subexpressions). Creating a container class like, for example, a class
7559 representing tensor products is more involved but this section should give
7560 you enough information so you can consult the source to GiNaC's predefined
7561 classes if you want to implement something more complicated.
7563 @subsection GiNaC's run-time type information system
7565 @cindex hierarchy of classes
7567 All algebraic classes (that is, all classes that can appear in expressions)
7568 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7569 @code{basic *} (which is essentially what an @code{ex} is) represents a
7570 generic pointer to an algebraic class. Occasionally it is necessary to find
7571 out what the class of an object pointed to by a @code{basic *} really is.
7572 Also, for the unarchiving of expressions it must be possible to find the
7573 @code{unarchive()} function of a class given the class name (as a string). A
7574 system that provides this kind of information is called a run-time type
7575 information (RTTI) system. The C++ language provides such a thing (see the
7576 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7577 implements its own, simpler RTTI.
7579 The RTTI in GiNaC is based on two mechanisms:
7584 The @code{basic} class declares a member variable @code{tinfo_key} which
7585 holds an unsigned integer that identifies the object's class. These numbers
7586 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7587 classes. They all start with @code{TINFO_}.
7590 By means of some clever tricks with static members, GiNaC maintains a list
7591 of information for all classes derived from @code{basic}. The information
7592 available includes the class names, the @code{tinfo_key}s, and pointers
7593 to the unarchiving functions. This class registry is defined in the
7594 @file{registrar.h} header file.
7598 The disadvantage of this proprietary RTTI implementation is that there's
7599 a little more to do when implementing new classes (C++'s RTTI works more
7600 or less automatically) but don't worry, most of the work is simplified by
7603 @subsection A minimalistic example
7605 Now we will start implementing a new class @code{mystring} that allows
7606 placing character strings in algebraic expressions (this is not very useful,
7607 but it's just an example). This class will be a direct subclass of
7608 @code{basic}. You can use this sample implementation as a starting point
7609 for your own classes.
7611 The code snippets given here assume that you have included some header files
7617 #include <stdexcept>
7618 using namespace std;
7620 #include <ginac/ginac.h>
7621 using namespace GiNaC;
7624 The first thing we have to do is to define a @code{tinfo_key} for our new
7625 class. This can be any arbitrary unsigned number that is not already taken
7626 by one of the existing classes but it's better to come up with something
7627 that is unlikely to clash with keys that might be added in the future. The
7628 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7629 which is not a requirement but we are going to stick with this scheme:
7632 const unsigned TINFO_mystring = 0x42420001U;
7635 Now we can write down the class declaration. The class stores a C++
7636 @code{string} and the user shall be able to construct a @code{mystring}
7637 object from a C or C++ string:
7640 class mystring : public basic
7642 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7645 mystring(const string &s);
7646 mystring(const char *s);
7652 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7655 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7656 macros are defined in @file{registrar.h}. They take the name of the class
7657 and its direct superclass as arguments and insert all required declarations
7658 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7659 the first line after the opening brace of the class definition. The
7660 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7661 source (at global scope, of course, not inside a function).
7663 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7664 declarations of the default constructor and a couple of other functions that
7665 are required. It also defines a type @code{inherited} which refers to the
7666 superclass so you don't have to modify your code every time you shuffle around
7667 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7668 class with the GiNaC RTTI (there is also a
7669 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7670 options for the class, and which we will be using instead in a few minutes).
7672 Now there are seven member functions we have to implement to get a working
7678 @code{mystring()}, the default constructor.
7681 @code{void archive(archive_node &n)}, the archiving function. This stores all
7682 information needed to reconstruct an object of this class inside an
7683 @code{archive_node}.
7686 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7687 constructor. This constructs an instance of the class from the information
7688 found in an @code{archive_node}.
7691 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7692 unarchiving function. It constructs a new instance by calling the unarchiving
7696 @cindex @code{compare_same_type()}
7697 @code{int compare_same_type(const basic &other)}, which is used internally
7698 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7699 -1, depending on the relative order of this object and the @code{other}
7700 object. If it returns 0, the objects are considered equal.
7701 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7702 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7703 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7704 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7705 must provide a @code{compare_same_type()} function, even those representing
7706 objects for which no reasonable algebraic ordering relationship can be
7710 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7711 which are the two constructors we declared.
7715 Let's proceed step-by-step. The default constructor looks like this:
7718 mystring::mystring() : inherited(TINFO_mystring) @{@}
7721 The golden rule is that in all constructors you have to set the
7722 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7723 it will be set by the constructor of the superclass and all hell will break
7724 loose in the RTTI. For your convenience, the @code{basic} class provides
7725 a constructor that takes a @code{tinfo_key} value, which we are using here
7726 (remember that in our case @code{inherited == basic}). If the superclass
7727 didn't have such a constructor, we would have to set the @code{tinfo_key}
7728 to the right value manually.
7730 In the default constructor you should set all other member variables to
7731 reasonable default values (we don't need that here since our @code{str}
7732 member gets set to an empty string automatically).
7734 Next are the three functions for archiving. You have to implement them even
7735 if you don't plan to use archives, but the minimum required implementation
7736 is really simple. First, the archiving function:
7739 void mystring::archive(archive_node &n) const
7741 inherited::archive(n);
7742 n.add_string("string", str);
7746 The only thing that is really required is calling the @code{archive()}
7747 function of the superclass. Optionally, you can store all information you
7748 deem necessary for representing the object into the passed
7749 @code{archive_node}. We are just storing our string here. For more
7750 information on how the archiving works, consult the @file{archive.h} header
7753 The unarchiving constructor is basically the inverse of the archiving
7757 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7759 n.find_string("string", str);
7763 If you don't need archiving, just leave this function empty (but you must
7764 invoke the unarchiving constructor of the superclass). Note that we don't
7765 have to set the @code{tinfo_key} here because it is done automatically
7766 by the unarchiving constructor of the @code{basic} class.
7768 Finally, the unarchiving function:
7771 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7773 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7777 You don't have to understand how exactly this works. Just copy these
7778 four lines into your code literally (replacing the class name, of
7779 course). It calls the unarchiving constructor of the class and unless
7780 you are doing something very special (like matching @code{archive_node}s
7781 to global objects) you don't need a different implementation. For those
7782 who are interested: setting the @code{dynallocated} flag puts the object
7783 under the control of GiNaC's garbage collection. It will get deleted
7784 automatically once it is no longer referenced.
7786 Our @code{compare_same_type()} function uses a provided function to compare
7790 int mystring::compare_same_type(const basic &other) const
7792 const mystring &o = static_cast<const mystring &>(other);
7793 int cmpval = str.compare(o.str);
7796 else if (cmpval < 0)
7803 Although this function takes a @code{basic &}, it will always be a reference
7804 to an object of exactly the same class (objects of different classes are not
7805 comparable), so the cast is safe. If this function returns 0, the two objects
7806 are considered equal (in the sense that @math{A-B=0}), so you should compare
7807 all relevant member variables.
7809 Now the only thing missing is our two new constructors:
7812 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7813 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7816 No surprises here. We set the @code{str} member from the argument and
7817 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7819 That's it! We now have a minimal working GiNaC class that can store
7820 strings in algebraic expressions. Let's confirm that the RTTI works:
7823 ex e = mystring("Hello, world!");
7824 cout << is_a<mystring>(e) << endl;
7827 cout << e.bp->class_name() << endl;
7831 Obviously it does. Let's see what the expression @code{e} looks like:
7835 // -> [mystring object]
7838 Hm, not exactly what we expect, but of course the @code{mystring} class
7839 doesn't yet know how to print itself. This can be done either by implementing
7840 the @code{print()} member function, or, preferably, by specifying a
7841 @code{print_func<>()} class option. Let's say that we want to print the string
7842 surrounded by double quotes:
7845 class mystring : public basic
7849 void do_print(const print_context &c, unsigned level = 0) const;
7853 void mystring::do_print(const print_context &c, unsigned level) const
7855 // print_context::s is a reference to an ostream
7856 c.s << '\"' << str << '\"';
7860 The @code{level} argument is only required for container classes to
7861 correctly parenthesize the output.
7863 Now we need to tell GiNaC that @code{mystring} objects should use the
7864 @code{do_print()} member function for printing themselves. For this, we
7868 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7874 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7875 print_func<print_context>(&mystring::do_print))
7878 Let's try again to print the expression:
7882 // -> "Hello, world!"
7885 Much better. If we wanted to have @code{mystring} objects displayed in a
7886 different way depending on the output format (default, LaTeX, etc.), we
7887 would have supplied multiple @code{print_func<>()} options with different
7888 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7889 separated by dots. This is similar to the way options are specified for
7890 symbolic functions. @xref{Printing}, for a more in-depth description of the
7891 way expression output is implemented in GiNaC.
7893 The @code{mystring} class can be used in arbitrary expressions:
7896 e += mystring("GiNaC rulez");
7898 // -> "GiNaC rulez"+"Hello, world!"
7901 (GiNaC's automatic term reordering is in effect here), or even
7904 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7906 // -> "One string"^(2*sin(-"Another string"+Pi))
7909 Whether this makes sense is debatable but remember that this is only an
7910 example. At least it allows you to implement your own symbolic algorithms
7913 Note that GiNaC's algebraic rules remain unchanged:
7916 e = mystring("Wow") * mystring("Wow");
7920 e = pow(mystring("First")-mystring("Second"), 2);
7921 cout << e.expand() << endl;
7922 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7925 There's no way to, for example, make GiNaC's @code{add} class perform string
7926 concatenation. You would have to implement this yourself.
7928 @subsection Automatic evaluation
7931 @cindex @code{eval()}
7932 @cindex @code{hold()}
7933 When dealing with objects that are just a little more complicated than the
7934 simple string objects we have implemented, chances are that you will want to
7935 have some automatic simplifications or canonicalizations performed on them.
7936 This is done in the evaluation member function @code{eval()}. Let's say that
7937 we wanted all strings automatically converted to lowercase with
7938 non-alphabetic characters stripped, and empty strings removed:
7941 class mystring : public basic
7945 ex eval(int level = 0) const;
7949 ex mystring::eval(int level) const
7952 for (int i=0; i<str.length(); i++) @{
7954 if (c >= 'A' && c <= 'Z')
7955 new_str += tolower(c);
7956 else if (c >= 'a' && c <= 'z')
7960 if (new_str.length() == 0)
7963 return mystring(new_str).hold();
7967 The @code{level} argument is used to limit the recursion depth of the
7968 evaluation. We don't have any subexpressions in the @code{mystring}
7969 class so we are not concerned with this. If we had, we would call the
7970 @code{eval()} functions of the subexpressions with @code{level - 1} as
7971 the argument if @code{level != 1}. The @code{hold()} member function
7972 sets a flag in the object that prevents further evaluation. Otherwise
7973 we might end up in an endless loop. When you want to return the object
7974 unmodified, use @code{return this->hold();}.
7976 Let's confirm that it works:
7979 ex e = mystring("Hello, world!") + mystring("!?#");
7983 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
7988 @subsection Optional member functions
7990 We have implemented only a small set of member functions to make the class
7991 work in the GiNaC framework. There are two functions that are not strictly
7992 required but will make operations with objects of the class more efficient:
7994 @cindex @code{calchash()}
7995 @cindex @code{is_equal_same_type()}
7997 unsigned calchash() const;
7998 bool is_equal_same_type(const basic &other) const;
8001 The @code{calchash()} method returns an @code{unsigned} hash value for the
8002 object which will allow GiNaC to compare and canonicalize expressions much
8003 more efficiently. You should consult the implementation of some of the built-in
8004 GiNaC classes for examples of hash functions. The default implementation of
8005 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8006 class and all subexpressions that are accessible via @code{op()}.
8008 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8009 tests for equality without establishing an ordering relation, which is often
8010 faster. The default implementation of @code{is_equal_same_type()} just calls
8011 @code{compare_same_type()} and tests its result for zero.
8013 @subsection Other member functions
8015 For a real algebraic class, there are probably some more functions that you
8016 might want to provide:
8019 bool info(unsigned inf) const;
8020 ex evalf(int level = 0) const;
8021 ex series(const relational & r, int order, unsigned options = 0) const;
8022 ex derivative(const symbol & s) const;
8025 If your class stores sub-expressions (see the scalar product example in the
8026 previous section) you will probably want to override
8028 @cindex @code{let_op()}
8031 ex op(size_t i) const;
8032 ex & let_op(size_t i);
8033 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8034 ex map(map_function & f) const;
8037 @code{let_op()} is a variant of @code{op()} that allows write access. The
8038 default implementations of @code{subs()} and @code{map()} use it, so you have
8039 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8041 You can, of course, also add your own new member functions. Remember
8042 that the RTTI may be used to get information about what kinds of objects
8043 you are dealing with (the position in the class hierarchy) and that you
8044 can always extract the bare object from an @code{ex} by stripping the
8045 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8046 should become a need.
8048 That's it. May the source be with you!
8051 @node A Comparison With Other CAS, Advantages, Adding classes, Top
8052 @c node-name, next, previous, up
8053 @chapter A Comparison With Other CAS
8056 This chapter will give you some information on how GiNaC compares to
8057 other, traditional Computer Algebra Systems, like @emph{Maple},
8058 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8059 disadvantages over these systems.
8062 * Advantages:: Strengths of the GiNaC approach.
8063 * Disadvantages:: Weaknesses of the GiNaC approach.
8064 * Why C++?:: Attractiveness of C++.
8067 @node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
8068 @c node-name, next, previous, up
8071 GiNaC has several advantages over traditional Computer
8072 Algebra Systems, like
8077 familiar language: all common CAS implement their own proprietary
8078 grammar which you have to learn first (and maybe learn again when your
8079 vendor decides to `enhance' it). With GiNaC you can write your program
8080 in common C++, which is standardized.
8084 structured data types: you can build up structured data types using
8085 @code{struct}s or @code{class}es together with STL features instead of
8086 using unnamed lists of lists of lists.
8089 strongly typed: in CAS, you usually have only one kind of variables
8090 which can hold contents of an arbitrary type. This 4GL like feature is
8091 nice for novice programmers, but dangerous.
8094 development tools: powerful development tools exist for C++, like fancy
8095 editors (e.g. with automatic indentation and syntax highlighting),
8096 debuggers, visualization tools, documentation generators@dots{}
8099 modularization: C++ programs can easily be split into modules by
8100 separating interface and implementation.
8103 price: GiNaC is distributed under the GNU Public License which means
8104 that it is free and available with source code. And there are excellent
8105 C++-compilers for free, too.
8108 extendable: you can add your own classes to GiNaC, thus extending it on
8109 a very low level. Compare this to a traditional CAS that you can
8110 usually only extend on a high level by writing in the language defined
8111 by the parser. In particular, it turns out to be almost impossible to
8112 fix bugs in a traditional system.
8115 multiple interfaces: Though real GiNaC programs have to be written in
8116 some editor, then be compiled, linked and executed, there are more ways
8117 to work with the GiNaC engine. Many people want to play with
8118 expressions interactively, as in traditional CASs. Currently, two such
8119 windows into GiNaC have been implemented and many more are possible: the
8120 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8121 types to a command line and second, as a more consistent approach, an
8122 interactive interface to the Cint C++ interpreter has been put together
8123 (called GiNaC-cint) that allows an interactive scripting interface
8124 consistent with the C++ language. It is available from the usual GiNaC
8128 seamless integration: it is somewhere between difficult and impossible
8129 to call CAS functions from within a program written in C++ or any other
8130 programming language and vice versa. With GiNaC, your symbolic routines
8131 are part of your program. You can easily call third party libraries,
8132 e.g. for numerical evaluation or graphical interaction. All other
8133 approaches are much more cumbersome: they range from simply ignoring the
8134 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8135 system (i.e. @emph{Yacas}).
8138 efficiency: often large parts of a program do not need symbolic
8139 calculations at all. Why use large integers for loop variables or
8140 arbitrary precision arithmetics where @code{int} and @code{double} are
8141 sufficient? For pure symbolic applications, GiNaC is comparable in
8142 speed with other CAS.
8147 @node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
8148 @c node-name, next, previous, up
8149 @section Disadvantages
8151 Of course it also has some disadvantages:
8156 advanced features: GiNaC cannot compete with a program like
8157 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8158 which grows since 1981 by the work of dozens of programmers, with
8159 respect to mathematical features. Integration, factorization,
8160 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8161 not planned for the near future).
8164 portability: While the GiNaC library itself is designed to avoid any
8165 platform dependent features (it should compile on any ANSI compliant C++
8166 compiler), the currently used version of the CLN library (fast large
8167 integer and arbitrary precision arithmetics) can only by compiled
8168 without hassle on systems with the C++ compiler from the GNU Compiler
8169 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8170 macros to let the compiler gather all static initializations, which
8171 works for GNU C++ only. Feel free to contact the authors in case you
8172 really believe that you need to use a different compiler. We have
8173 occasionally used other compilers and may be able to give you advice.}
8174 GiNaC uses recent language features like explicit constructors, mutable
8175 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8176 literally. Recent GCC versions starting at 2.95.3, although itself not
8177 yet ANSI compliant, support all needed features.
8182 @node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
8183 @c node-name, next, previous, up
8186 Why did we choose to implement GiNaC in C++ instead of Java or any other
8187 language? C++ is not perfect: type checking is not strict (casting is
8188 possible), separation between interface and implementation is not
8189 complete, object oriented design is not enforced. The main reason is
8190 the often scolded feature of operator overloading in C++. While it may
8191 be true that operating on classes with a @code{+} operator is rarely
8192 meaningful, it is perfectly suited for algebraic expressions. Writing
8193 @math{3x+5y} as @code{3*x+5*y} instead of
8194 @code{x.times(3).plus(y.times(5))} looks much more natural.
8195 Furthermore, the main developers are more familiar with C++ than with
8196 any other programming language.
8199 @node Internal Structures, Expressions are reference counted, Why C++? , Top
8200 @c node-name, next, previous, up
8201 @appendix Internal Structures
8204 * Expressions are reference counted::
8205 * Internal representation of products and sums::
8208 @node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
8209 @c node-name, next, previous, up
8210 @appendixsection Expressions are reference counted
8212 @cindex reference counting
8213 @cindex copy-on-write
8214 @cindex garbage collection
8215 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8216 where the counter belongs to the algebraic objects derived from class
8217 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8218 which @code{ex} contains an instance. If you understood that, you can safely
8219 skip the rest of this passage.
8221 Expressions are extremely light-weight since internally they work like
8222 handles to the actual representation. They really hold nothing more
8223 than a pointer to some other object. What this means in practice is
8224 that whenever you create two @code{ex} and set the second equal to the
8225 first no copying process is involved. Instead, the copying takes place
8226 as soon as you try to change the second. Consider the simple sequence
8231 #include <ginac/ginac.h>
8232 using namespace std;
8233 using namespace GiNaC;
8237 symbol x("x"), y("y"), z("z");
8240 e1 = sin(x + 2*y) + 3*z + 41;
8241 e2 = e1; // e2 points to same object as e1
8242 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8243 e2 += 1; // e2 is copied into a new object
8244 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8248 The line @code{e2 = e1;} creates a second expression pointing to the
8249 object held already by @code{e1}. The time involved for this operation
8250 is therefore constant, no matter how large @code{e1} was. Actual
8251 copying, however, must take place in the line @code{e2 += 1;} because
8252 @code{e1} and @code{e2} are not handles for the same object any more.
8253 This concept is called @dfn{copy-on-write semantics}. It increases
8254 performance considerably whenever one object occurs multiple times and
8255 represents a simple garbage collection scheme because when an @code{ex}
8256 runs out of scope its destructor checks whether other expressions handle
8257 the object it points to too and deletes the object from memory if that
8258 turns out not to be the case. A slightly less trivial example of
8259 differentiation using the chain-rule should make clear how powerful this
8264 symbol x("x"), y("y");
8268 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8269 cout << e1 << endl // prints x+3*y
8270 << e2 << endl // prints (x+3*y)^3
8271 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8275 Here, @code{e1} will actually be referenced three times while @code{e2}
8276 will be referenced two times. When the power of an expression is built,
8277 that expression needs not be copied. Likewise, since the derivative of
8278 a power of an expression can be easily expressed in terms of that
8279 expression, no copying of @code{e1} is involved when @code{e3} is
8280 constructed. So, when @code{e3} is constructed it will print as
8281 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8282 holds a reference to @code{e2} and the factor in front is just
8285 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8286 semantics. When you insert an expression into a second expression, the
8287 result behaves exactly as if the contents of the first expression were
8288 inserted. But it may be useful to remember that this is not what
8289 happens. Knowing this will enable you to write much more efficient
8290 code. If you still have an uncertain feeling with copy-on-write
8291 semantics, we recommend you have a look at the
8292 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8293 Marshall Cline. Chapter 16 covers this issue and presents an
8294 implementation which is pretty close to the one in GiNaC.
8297 @node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
8298 @c node-name, next, previous, up
8299 @appendixsection Internal representation of products and sums
8301 @cindex representation
8304 @cindex @code{power}
8305 Although it should be completely transparent for the user of
8306 GiNaC a short discussion of this topic helps to understand the sources
8307 and also explain performance to a large degree. Consider the
8308 unexpanded symbolic expression
8310 $2d^3 \left( 4a + 5b - 3 \right)$
8313 @math{2*d^3*(4*a+5*b-3)}
8315 which could naively be represented by a tree of linear containers for
8316 addition and multiplication, one container for exponentiation with base
8317 and exponent and some atomic leaves of symbols and numbers in this
8322 @cindex pair-wise representation
8323 However, doing so results in a rather deeply nested tree which will
8324 quickly become inefficient to manipulate. We can improve on this by
8325 representing the sum as a sequence of terms, each one being a pair of a
8326 purely numeric multiplicative coefficient and its rest. In the same
8327 spirit we can store the multiplication as a sequence of terms, each
8328 having a numeric exponent and a possibly complicated base, the tree
8329 becomes much more flat:
8333 The number @code{3} above the symbol @code{d} shows that @code{mul}
8334 objects are treated similarly where the coefficients are interpreted as
8335 @emph{exponents} now. Addition of sums of terms or multiplication of
8336 products with numerical exponents can be coded to be very efficient with
8337 such a pair-wise representation. Internally, this handling is performed
8338 by most CAS in this way. It typically speeds up manipulations by an
8339 order of magnitude. The overall multiplicative factor @code{2} and the
8340 additive term @code{-3} look somewhat out of place in this
8341 representation, however, since they are still carrying a trivial
8342 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8343 this is avoided by adding a field that carries an overall numeric
8344 coefficient. This results in the realistic picture of internal
8347 $2d^3 \left( 4a + 5b - 3 \right)$:
8350 @math{2*d^3*(4*a+5*b-3)}:
8356 This also allows for a better handling of numeric radicals, since
8357 @code{sqrt(2)} can now be carried along calculations. Now it should be
8358 clear, why both classes @code{add} and @code{mul} are derived from the
8359 same abstract class: the data representation is the same, only the
8360 semantics differs. In the class hierarchy, methods for polynomial
8361 expansion and the like are reimplemented for @code{add} and @code{mul},
8362 but the data structure is inherited from @code{expairseq}.
8365 @node Package Tools, ginac-config, Internal representation of products and sums, Top
8366 @c node-name, next, previous, up
8367 @appendix Package Tools
8369 If you are creating a software package that uses the GiNaC library,
8370 setting the correct command line options for the compiler and linker
8371 can be difficult. GiNaC includes two tools to make this process easier.
8374 * ginac-config:: A shell script to detect compiler and linker flags.
8375 * AM_PATH_GINAC:: Macro for GNU automake.
8379 @node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
8380 @c node-name, next, previous, up
8381 @section @command{ginac-config}
8382 @cindex ginac-config
8384 @command{ginac-config} is a shell script that you can use to determine
8385 the compiler and linker command line options required to compile and
8386 link a program with the GiNaC library.
8388 @command{ginac-config} takes the following flags:
8392 Prints out the version of GiNaC installed.
8394 Prints '-I' flags pointing to the installed header files.
8396 Prints out the linker flags necessary to link a program against GiNaC.
8397 @item --prefix[=@var{PREFIX}]
8398 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8399 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8400 Otherwise, prints out the configured value of @env{$prefix}.
8401 @item --exec-prefix[=@var{PREFIX}]
8402 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8403 Otherwise, prints out the configured value of @env{$exec_prefix}.
8406 Typically, @command{ginac-config} will be used within a configure
8407 script, as described below. It, however, can also be used directly from
8408 the command line using backquotes to compile a simple program. For
8412 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8415 This command line might expand to (for example):
8418 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8419 -lginac -lcln -lstdc++
8422 Not only is the form using @command{ginac-config} easier to type, it will
8423 work on any system, no matter how GiNaC was configured.
8426 @node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
8427 @c node-name, next, previous, up
8428 @section @samp{AM_PATH_GINAC}
8429 @cindex AM_PATH_GINAC
8431 For packages configured using GNU automake, GiNaC also provides
8432 a macro to automate the process of checking for GiNaC.
8435 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8436 [, @var{ACTION-IF-NOT-FOUND}]]])
8444 Determines the location of GiNaC using @command{ginac-config}, which is
8445 either found in the user's path, or from the environment variable
8446 @env{GINACLIB_CONFIG}.
8449 Tests the installed libraries to make sure that their version
8450 is later than @var{MINIMUM-VERSION}. (A default version will be used
8454 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8455 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8456 variable to the output of @command{ginac-config --libs}, and calls
8457 @samp{AC_SUBST()} for these variables so they can be used in generated
8458 makefiles, and then executes @var{ACTION-IF-FOUND}.
8461 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8462 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8466 This macro is in file @file{ginac.m4} which is installed in
8467 @file{$datadir/aclocal}. Note that if automake was installed with a
8468 different @samp{--prefix} than GiNaC, you will either have to manually
8469 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8470 aclocal the @samp{-I} option when running it.
8473 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8474 * Example package:: Example of a package using AM_PATH_GINAC.
8478 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8479 @c node-name, next, previous, up
8480 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8482 Simply make sure that @command{ginac-config} is in your path, and run
8483 the configure script.
8490 The directory where the GiNaC libraries are installed needs
8491 to be found by your system's dynamic linker.
8493 This is generally done by
8496 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8502 setting the environment variable @env{LD_LIBRARY_PATH},
8505 or, as a last resort,
8508 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8509 running configure, for instance:
8512 LDFLAGS=-R/home/cbauer/lib ./configure
8517 You can also specify a @command{ginac-config} not in your path by
8518 setting the @env{GINACLIB_CONFIG} environment variable to the
8519 name of the executable
8522 If you move the GiNaC package from its installed location,
8523 you will either need to modify @command{ginac-config} script
8524 manually to point to the new location or rebuild GiNaC.
8535 --with-ginac-prefix=@var{PREFIX}
8536 --with-ginac-exec-prefix=@var{PREFIX}
8539 are provided to override the prefix and exec-prefix that were stored
8540 in the @command{ginac-config} shell script by GiNaC's configure. You are
8541 generally better off configuring GiNaC with the right path to begin with.
8545 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8546 @c node-name, next, previous, up
8547 @subsection Example of a package using @samp{AM_PATH_GINAC}
8549 The following shows how to build a simple package using automake
8550 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8554 #include <ginac/ginac.h>
8558 GiNaC::symbol x("x");
8559 GiNaC::ex a = GiNaC::sin(x);
8560 std::cout << "Derivative of " << a
8561 << " is " << a.diff(x) << std::endl;
8566 You should first read the introductory portions of the automake
8567 Manual, if you are not already familiar with it.
8569 Two files are needed, @file{configure.in}, which is used to build the
8573 dnl Process this file with autoconf to produce a configure script.
8575 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8581 AM_PATH_GINAC(0.9.0, [
8582 LIBS="$LIBS $GINACLIB_LIBS"
8583 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8584 ], AC_MSG_ERROR([need to have GiNaC installed]))
8589 The only command in this which is not standard for automake
8590 is the @samp{AM_PATH_GINAC} macro.
8592 That command does the following: If a GiNaC version greater or equal
8593 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8594 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8595 the error message `need to have GiNaC installed'
8597 And the @file{Makefile.am}, which will be used to build the Makefile.
8600 ## Process this file with automake to produce Makefile.in
8601 bin_PROGRAMS = simple
8602 simple_SOURCES = simple.cpp
8605 This @file{Makefile.am}, says that we are building a single executable,
8606 from a single source file @file{simple.cpp}. Since every program
8607 we are building uses GiNaC we simply added the GiNaC options
8608 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8609 want to specify them on a per-program basis: for instance by
8613 simple_LDADD = $(GINACLIB_LIBS)
8614 INCLUDES = $(GINACLIB_CPPFLAGS)
8617 to the @file{Makefile.am}.
8619 To try this example out, create a new directory and add the three
8622 Now execute the following commands:
8625 $ automake --add-missing
8630 You now have a package that can be built in the normal fashion
8639 @node Bibliography, Concept Index, Example package, Top
8640 @c node-name, next, previous, up
8641 @appendix Bibliography
8646 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8649 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8652 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8655 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8658 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8659 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8662 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8663 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8664 Academic Press, London
8667 @cite{Computer Algebra Systems - A Practical Guide},
8668 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8671 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8672 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8675 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8676 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8679 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8684 @node Concept Index, , Bibliography, Top
8685 @c node-name, next, previous, up
8686 @unnumbered Concept Index